Reflectarray Antennas: Theory, Designs, and Applications 1118846761, 9781118846766

Table of contents :
Cover
Half Title
Reflectarray Antennas: Theory, Designs,
and Applications
© 2018
Dedication
Contents
Foreword
Preface
Acknowledgments
1 Introduction to Reflectarray Antennas
2 Analysis and Design of Reflectarray Elements
3 System Design and Aperture Efficiency Analysis
4 Radiation Analysis Techniques
5 Bandwidth of Reflectarray Antennas
6 Reflectarray Design Examples
7 Broadband and Multiband Reflectarray Antennas
8 Terahertz, Infrared, and Optical Reflectarray Antennas
9 Multi‐Beam and Shaped‐Beam Reflectarray Antennas
10 Beam‐Scanning Reflectarray Antennas
11 Reflectarray Engineering and Emerging Applications
Index

Citation preview

Reflectarray Antennas: Theory, Designs, and Applications

Reflectarray Antennas: Theory, Designs, and Applications Payam Nayeri

Colorado School of Mines USA

Fan Yang

Tsinghua University China

Atef Z. Elsherbeni

Colorado School of Mines USA



This edition first published 2018 © 2018 John Wiley & Sons Ltd All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions. The right of Payam Nayeri, Fan Yang, and Atef Z. Elsherbeni to be identified as the authors of this work has been asserted in accordance with law. Registered Offices John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK Editorial Office The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Wiley also publishes its books in a variety of electronic formats and by print‐on‐demand. Some content that appears in standard print versions of this book may not be available in other formats. Limit of Liability/Disclaimer of Warranty While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Library of Congress Cataloging‐in‐Publication data applied for ISBN: 978‐1‐118‐84676‐6 Cover design by Wiley Cover images: (Background) © Andrey Prokhorov/Gettyimages; (Foreground) Courtesy of Payam Nayeri, Fan Yang, and Atef Z. Elsherbeni Set in 10/12pt Warnock by SPi Global, Pondicherry, India 10 9 8 7 6 5 4 3 2 1

To my parents who I am eternally grateful for their love, support, and encouragement throughout my career Payam Nayeri To my colleagues and students, and to my family

Fan Yang

To my wife, Magda, daughters, Dalia and Donia, son, Tamer, and the memory of my parents Atef Z. Elsherbeni

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Contents Foreword  xiii Preface  xv Acknowledgments  xvii 1 Introduction to Reflectarray Antennas  1

1.1 Reflectarray Concept  1 1.2 Reflectarray Developments  2 1.3 Overview of this Book  5 ­References  7

2 Analysis and Design of Reflectarray Elements  9

2.1 Phase‐Shift Distribution on the Reflectarray Aperture  9 2.2 Phase Tuning Approaches for Reflectarray Elements  13 2.2.1 Elements with Phase/Time‐Delay Lines  14 2.2.2 Elements with Variable Sizes  15 2.2.3 Elements with Variable Rotation Angles  16 2.3 Element Analysis Methods  18 2.3.1 Periodic Boundary Conditions and Floquet Port Excitation  19 2.3.2 Metallic Waveguide Simulators  19 2.3.3 Analytical Circuit Models  21 2.3.4 Comparison of Element Analysis Techniques  22 2.3.4.1 Comparison between PBC and Metallic Waveguides  23 2.3.4.2 Comparison between PBC and the Circuit Model  24 2.4 Examples of Classic Reflectarray Elements  26 2.4.1 Rectangular Patch with Phase‐Delay Lines  26 2.4.2 Variable Size Square Patch  30 2.4.3 Single Slot Ring Elements  33 2.5 Reflectarray Element Characteristics and Design Considerations  37 2.5.1 Frequency Behavior of Element Reflection Coefficients  37 2.5.2 Effects of Oblique Incidence Angles on Element Reflection Coefficients  37 2.5.3 Sources of Phase Error in Reflectarray Element Design  41 2.6 Reflectarray Element Measurements  43 ­References  46

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3 System Design and Aperture Efficiency Analysis  49 3.1 A General Feed Model  49 3.1.1 Models of Linearly Polarized and Circularly Polarized Feeds  50 3.1.2 Balanced Feed Models  51 3.2 Aperture Efficiency  53 3.2.1 Spillover Efficiency  53 3.2.2 Illumination Efficiency  54 3.2.3 Effects of Aperture Shape on Efficiency  55 3.2.4 Effects of Feed Location on Efficiency  59 3.3 Aperture Blockage and Edge Diffraction  60 3.3.1 Aperture Blockage and Offset Systems  60 3.3.2 Edge Taper and Edge Diffraction  63 3.4 The Analogy between a Reflectarray and a Parabolic Reflector  70 3.4.1 The Offset System Configurations  71 3.4.2 Analogous Offset Reflector  72 3.4.2.1 Transformation from Reflector to Reflectarray System  72 3.4.2.2 Transformation from Reflectarray to Reflector System  75 3.4.3 Example of Analogous Offset Systems  76 ­References  77 4 Radiation Analysis Techniques  79 4.1 Array Theory Approach: The Robust Analysis Technique  80 4.1.1 Idealized Feed and Element Patterns  80 4.1.2 Element Excitations and Reflectarray Radiation Pattern  81 4.2 Aperture Field Approach: The Classical Analysis Technique  82 4.2.1 Complex Feed Patterns  82 4.2.2 Field Transformations from Feed to Aperture and Equivalent Surface Current  83 4.2.3 Near‐Field to Far‐Field Transforms and Reflectarray Radiation Pattern  85 4.3 Important Topics in Reflectarray Radiation Analysis  87 4.3.1 Principal Radiation Planes  87 4.3.2 Co‐ and Cross‐Polarized Patterns  89 4.3.3 Antenna Directivity  90 4.3.4 Antenna Efficiency and Gain  91 4.3.5 Spectral Transforms and Computational Speedup  94 4.4 Full‐Wave Simulation Approaches  96 4.4.1 Constructed Aperture Currents Under Local‐Periodicity Approximation  96 4.4.2 Complete Reflectarray Models  96 4.5 Numerical Examples  98 4.5.1 Comparison of the Array Theory and Aperture Field Analysis Techniques  98 4.5.1.1 Example 1: Reflectarray Antenna with a Broadside Beam  99 4.5.1.2 Example 2: Reflectarray Antenna with an Off‐Broadside Beam  100

Contents

4.5.1.3 Comparison of Calculated Directivity versus Frequency  103 4.5.2 Consideration in the Array Theory Technique: Element Pattern Effect  105 4.5.3 Consideration in the Aperture Field Technique: Variations of Equivalence Principle  106 4.5.4 Comparisons with Full‐Wave Technique  107 ­References  110 5 Bandwidth of Reflectarray Antennas  113

5.1 Bandwidth Constraints in Reflectarray Antennas  113 5.1.1 Frequency Behavior of Element Phase Error  113 5.1.2 Frequency Behavior of Spatial Phase Delay  115 5.1.3 Aperture Phase Error and Reflectarray Bandwidth Limitations  118 5.2 Reflectarray Element Bandwidth  121 5.2.1 Physics of Element Bandwidth Constraints  121 5.2.2 Parametric Studies on Element Bandwidth  122 5.3 Reflectarray System Bandwidth  135 5.3.1 Effect of Aperture Size on Reflectarray Bandwidth  135 5.3.2 Effects of Element on Reflectarray Bandwidth  140 ­References  144

6 Reflectarray Design Examples  147

6.1 A Ku‐band Reflectarray Antenna: A Step‐by‐Step Design Example  147 6.1.1 Feed Antenna Characteristics  147 6.1.2 Reflectarray System Design  150 6.1.3 Reflectarray Element Design  153 6.1.4 Radiation Analysis  156 6.1.5 Fabrication and Measurements  159 6.2 A Circularly Polarized Reflectarray Antenna using an Element Rotation Technique  165 6.3 Bandwidth Comparison of Reflectarray Designs using Different Elements  169 ­References  176

7 Broadband and Multiband Reflectarray Antennas  179

7.1 Broadband Reflectarray Design Topologies  179 7.1.1 Multilayer Multi‐Resonance Elements  179 7.1.2 Single‐Layer Multi‐Resonance Elements  181 7.1.3 Sub‐Wavelength Elements  184 7.1.4 Reflectarrays Employing Single‐Layer and Double‐Layer Sub‐Wavelength Elements  188 7.1.5 Broadband Design Methods for Large Reflectarrays  197 7.2 Phase Synthesis for Broadband Operation  197 7.2.1 A Phase Synthesized Broadband Reflectarray  200 7.2.2 A Dual‐Frequency Broadband Reflectarray  203

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7.3 Multiband Reflectarray Designs  206 7.3.1 A Single‐Layer Dual‐Band Circularly Polarized Reflectarray  210 7.3.2 A Single-Layer Tri-Band Reflectarray  213 ­References  221 8 Terahertz, Infrared, and Optical Reflectarray Antennas  227

8.1 Above Microwave Frequencies  227 8.2 Material Characteristics at Terahertz and Infrared Frequencies  228 8.2.1 Optical Measurements and Electromagnetic Parameters  228 8.2.2 Measured Properties of Conductors and Dielectric Materials  229 8.2.3 Calculating Drude Model Parameters for Conductors  229 8.3 Element Losses at Infrared Frequencies  234 8.3.1 Conductor Losses  234 8.3.1.1 Effect of Conductor Thickness  234 8.3.1.2 Effect of Complex Conductivity  237 8.3.2 Dielectric Losses  240 8.3.3 Effect of Losses on Reflection Properties of Elements  241 8.3.4 Circuit‐Model Analysis  242 8.3.4.1 Circuit Theory and Loss Study  242 8.3.4.2 Zero‐Pole Analysis of Element Performance  243 8.4 Reflectarray Design Methodologies and Enabling Technologies  245 8.4.1 Reflectarrays with Patch Elements  245 8.4.2 Dielectric Resonator Reflectarrays  248 8.4.3 Dielectric Reflectarrays  251 8.4.3.1 Dielectric Property and 3D Printing Technique  251 8.4.3.2 Dielectric Reflectarray Design  253 8.4.3.3 Dielectric Reflectarray Prototypes and Measurements  259 8.5 Future Trends  261 ­References  264

9 Multi‐Beam and Shaped‐Beam Reflectarray Antennas  267 9.1 Direct Design Approaches for Multi‐Beam Reflectarrays  268 9.1.1 Geometrical Aperture Division  268 9.1.2 Superposition of Aperture Fields  271 9.1.3 Comparison of Direct Design Approaches  272 9.2 Synthesis Design Approaches for Shaped‐ and Multi‐Beam Reflectarrays  275 9.2.1 Basics of Synthesis Techniques  275 9.2.2 Local‐Search Techniques  276 9.2.3 Global‐Search Techniques  279 9.2.4 Full‐Wave Optimization Design Approaches  280 9.3 Practical Reflectarray Designs  281 9.3.1 Single‐Feed Reflectarray with Multiple Symmetric Beams  281 9.3.2 Feed Reflectarrays with Multiple Asymmetric Beams  286 9.3.3 Shaped‐Beam Reflectarrays  294 9.3.4 Multi‐Feed Multi‐Beam Reflectarrays  297 ­References  300

Contents

10 Beam‐Scanning Reflectarray Antennas  303 10.1 Beam‐Scanning Approaches for Reflectarray Antennas  304 10.1.1 Design Methodologies  304 10.1.2 Classifications Based on Reflector Type  306 10.2 Feed‐Tuning Techniques  307 10.2.1 Fully Illuminated Single‐Reflector Configurations  307 10.2.1.1 Parabolic‐Phase Apertures  307 10.2.1.2 Non‐Parabolic‐Phase Apertures  313 10.2.2 Partially Illuminated Single‐Reflector Configurations  324 10.2.2.1 Parabolic Cylindrical‐Phase Reflectarray Antennas (PCPRA)  324 10.2.2.2 Parabolic Torus‐Phase Reflectarray Antennas (PTPRA)  329 10.2.2.3 Spherical‐Phase Reflectarray Antennas (SPRA)  331 10.2.3 Dual‐Reflector Configurations  334 10.2.3.1 Parabolic Reflector/Reflectarray Antennas  334 10.2.3.2 Non‐Parabolic Reflector/Reflectarray Antennas  336 10.2.4 Summary of Feed‐Tuning Techniques  337 10.3 Aperture Phase‐Tuning Techniques  339 10.3.1 Basics of Aperture Phase Tuning  339 10.3.2 Enabling Technologies  341 10.3.2.1 Mechanical Actuators/Motors  341 10.3.2.2 Electronic Devices  343 10.3.2.3 Functional Materials  352 10.4 Frontiers in Beam‐Scanning Reflectarray Research  355 10.4.1 Active Reflectarrays  355 10.4.2 Comparison Between Analog and Digital Phase Control  355 10.4.3 Sub‐Array Techniques  358 10.4.4 Hybrid Configurations  359 ­References  359 11 Reflectarray Engineering and Emerging Applications  367

11.1 Advanced Reflectarray Geometries  367 11.1.1 Conformal Reflectarrays  367 11.1.1.1 Analysis of Conformal Reflectarrays  367 11.1.1.2 Radiation Characteristics of Conformal Reflectarrays on Cylindrical Surfaces  369 11.1.2 Dual‐Reflectarrays 375 11.2 Reflectarrays for Satellite Applications  379 11.2.1 An L‐Band Reflectarray for the Beidou Satellite System  381 11.2.2 Reflectarrays Integrated with Solar Cells  384 11.3 Power Combining and Amplifying Reflectarrays  388 11.4 A Perspective on Reflectarray Antennas  393 11.4.1 Large‐Aperture Planar Reflectarray Antennas  393 11.4.2 Reflectarray Antennas with Broad Bandwidth, Beam‐Scanning Capability, and Low Cost  396 11.4.3 From Reflectarray Antennas to Transmitarray Antennas  396 ­References  397 Index  401

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Foreword Although the concept of the reflectarray antenna was first introduced in 1963, the vast interest in it did not come about until in the late 1980s with the development of low‐ profile microstrip antennas. From the word reflectarray, it can be deduced that this is an antenna that combines the unique features of a parabolic reflector and a phased array. Thus, a low‐profile reflectarray consists of an array of microstrip elements that are provided with a set of pre‐adjusted phases to form a focused beam when illuminated by a feed, in a similar way to a parabolic reflector. The array elements can be printed onto either a flat surface or a slightly curved surface and have been demonstrated to have the ability to produce a high‐gain pencil beam, a contour‐shaped beam, multiple beams, or an electronically scanned beam. Because the array elements in a reflectarray are not physically interconnected, it can produce a high‐gain beam with relatively high efficiency similar to that produced by a parabolic reflector. There were several pioneers that initiated the study of printed reflectarrays during the late 1980s. I thought about the idea of a reflectarray due to my earlier work experiences with microstrip antennas and frequency selective surfaces (FSS). At certain resonant frequencies, the FSS can only reflect as a nearly perfect conductor since all elements are identical. It cannot cause the reflected waves to form a phase‐coherent beam. However, if each FSS element is designed differently with appropriate phase delay, a coherent beam can then be formed and a printed reflectarray is consequently formed. This book gives a comprehensive presentation of reflectarray antennas. Chapter 1 is a general overview of the operating principles as well as the developmental history of reflectarray antennas. Chapters 2 through 5 provide very complete and detailed design and analysis techniques, including the important element characterization and selection, radiation efficiency analysis and system design, various radiation analysis approaches and tradeoffs, and the most critical bandwidth issues and analysis. Chapter 6 gives a few specific design examples; in particular, a Ku‐band step‐by‐step design example and a circularly polarized reflectarray design. It is well known that the bandwidth limitation generally presents critical issues in reflectarray design. Chapter 7 is devoted to broadband solutions by presenting several bandwidth widening techniques and multiband approaches. The Terahertz, infrared, and optical frequencies have been found to be the frontier of research and application for antennas. Reflectarray antennas have also found applications in these extremely high frequency areas and are presented in Chapter 8, where the critical issues of material characterization and element loss are discussed. Low‐loss dielectric resonators, used as elements, are also presented in this chapter. A single reflectarray antenna can not only be designed to produce a high‐gain

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pencil beam, but, due to its many array elements, also has the ability to generate a specifically contour‐shaped beam as well as multiple beams. Chapter 9 gives a thorough presentation of the design approaches, which include direct design approaches and synthesis design approaches for a single reflectarray to radiate a contour‐shaped beam or multiple beams. Chapter 10 engages in discussion about a reflectarray’s beam scanning capability and design approaches. One of the key advantages of the reflectarray is its ability to achieve fast electronic beam scanning by implanting a low‐loss phase shifter into each of its elements without the need for expensive transmit/receive modules and high‐loss power division network. Thus, the reflectarray, owing to the hybrid nature of reflector and array, can behave like an efficient high‐gain parabolic reflector and a relatively low‐cost phased array. Finally, Chapter 11 discusses several emerging and future applications of reflectarray antennas, such as a reflectarray conformally mounted on curved surfaces, satellite applications, integration with solar cells, amplifying reflectarrays, dual‐reflectarrays, very large aperture applications, and so on. By comparing this book with the very first reflectarray book published by the Wiley‐ IEEE Press (Huang and Encinar) in 2008, this book not only gives more updated information, but also gives more detailed analysis and design presentations. The authors of that 2008 book also presented their own pioneering contribution in the areas such as broadband design using sub‐wavelength patch elements, a special phase synthesis approach, and single as well as multilayer approaches. In particular, a single layer design with tri‐band circular polarization performance was achieved. It was a cooperative effort that fulfilled my contractual request from the Jet Propulsion Laboratory while the authors were teaching at the University of Mississippi. A unique split‐square ring element was also used to achieve excellent circular polarization for this single‐layer multiband reflectarray. In that book, the authors presented their own contributions in the area of Terahertz and infrared reflectarray applications. In addition, the synthesis technique for a single reflectarray to achieve multiple beams and specifically shaped beams was presented as well. The electronic beam scanning capability of the reflectarray was also fully discussed with several well‐presented new design approaches. This book is well organized and has significant amount of information in design and analysis with many practical application results augmented with adequate number of references to help the readers to comprehend. Undoubtedly, I believe this book is not only well suited as a university text book but also is an excellent source of design and analysis information for antenna engineers for many years to come. Dr. John Huang Principle Engineer, retiree of The Spacecraft antenna research group Jet Propulsion Laboratory California Institute of Technology Pasadena, California, USA

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Preface High‐gain antennas are an essential part of long‐distance wireless communications, radar, and remote sensing systems, which vary with frequency, coverage, resolution, and flexibility of operation. The conventional choices for antennas in these systems were typically reflectors, lenses, or arrays. In recent years, however, a new generation of high‐gain antennas has emerged that combines the favorable features of both printed arrays and reflector antennas and creates a high‐gain antenna with low‐profile, low‐mass, and low‐cost features. This antenna is known as the reflectarray. The reflectarray is an antenna with a flat reflecting surface consisting of hundreds of elements and an illuminating feed antenna. The hybrid nature of the reflectarray antenna offers more flexibility in aperture phase control and can provide advantages over both reflectors and array antennas for many applications. The elements of the reflectarray are individually designed to reflect the electromagnetic wave with a certain phase to compensate for the phase delay caused by the spatial feed. The phase shift of the elements is realized using various methods such as variable‐size elements. Single and multilayer reflectarrays have been designed to achieve broadband and multiband performance from microwave frequencies up to the THz range. Meanwhile, the direct control of the phase of every element in the array allows multi‐beam or shaped beam performance with single or multiple feeds. Another advantage of reflectarrays is the ability of the antenna to scan the main beam to large angles off broadside. The advantages of reflectarrays, such as being low‐profile, lightweight, and having conformal geometry, make it desirable for various communication systems, especially for mobile platforms. Its applications in space exploration, satellite communications, remote sensing, and radar systems are rising, and will continue to increase in the future. In addition, the current printed circuit board (PCB) fabrication technology and available low‐cost commercial laminates, allows for low‐cost rapid prototype fabrication. This is also leading to commercial implementation and large‐scale fabrication of reflectarray antennas. The potential of reflectarray capabilities has not yet been fully exploited. Researchers in this field are constantly presenting new ideas and designs ranging from advanced materials to multifunctional system designs. As such, it is expected that this field will remain an active area of research, and there is no doubt that reflectarrays will become an important member of the antenna family. The aim of the book is to provide scientists and engineers in the fields of antenna, microwave, and electromagnetics, with up‐to‐date knowledge of reflectarray antenna theories, designs, and applications. This book will provide the reader with an overview of the reflectarray antenna research history and state‐of‐the‐art, good knowledge of the

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basic theories for design and analysis of reflectarray antennas, and detailed design procedures for a wide range of diversified and advanced applications. The prerequisite for this book is that the readers should be familiar with the basics of antenna engineering. The first part of this book includes the fundamental theories of reflectarrays, and is intended for engineers that know the basics of antenna theory and are becoming familiar with this new generation of high‐gain antennas. Chapter 1 introduces the reflectarray concept and historical backgrounds, and provides an overview of this book. Chapter  2 provides a comprehensive coverage of aperture phase requirements in reflectarray systems, phasing element design methodologies, and element analysis techniques. Reflectarray system design and efficiency analysis are introduced in Chapter 3. A detailed coverage of the various methods to compute the radiation characteristics of reflectarray antennas is presented in Chapter 4. The bandwidth characteristics of reflectarray antennas are studied in detail in Chapter 5. A variety of reflectarray designs are presented in Chapter 6 that can serve as a useful reference for interested readers. The second part of the book is intended for researchers and specialists that have a good knowledge of the basic theories in reflectarrays, and aim to design reflectarray antennas for specific applications/operations. It starts with a comprehensive overview of broadband and multiband reflectarray antennas in Chapter 7. Reflectarrays operating above microwave frequencies such as in the terahertz, infrared, and optical spectrums are introduced in Chapter  8. A detailed coverage of multi‐beam and shaped‐beam reflectarrays is presented in Chapter 9. Chapter 10 presents beam‐scanning reflectarray antennas, where the extensive research on these types of reflectarrays is summarized and analyzed in a comprehensive fashion. The final chapter of this book, Chapter 11, is devoted to advanced reflectarray antenna configurations.

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Acknowledgments The work presented in this book was supported in part by the following institutions: National Aeronautics and Space Administration (NASA) National Science Foundation (NSF) Tsinghua National Laboratory for Information Science and Technology Chinese High‐Technology Research and Development Program (863‐Program) The authors would like to thank the technical reviewers for their insightful feedback which enhanced the clarity and efficacy of this work. We would also like to express our sincere gratitude to our colleagues and students over these years: Dr. John Huang, Prof. Yahya Rahmat‐Samii, Prof. Jianhua Lu, Prof. Shenheng Xu, Prof. Glenn Boreman, Prof. Hao Xin, Prof. Maokun Li, Dr. Ang Yu, Dr. Chye Hwa Loo, Dr. Wenxing An, Dr. Yilin Mao, Dr. Ahmed Hassan Abdelrahman, Dr. Huanhuan Yang, Dr. Ruyuan Deng, Mr. Yanghyo Kim, Ms. Bhavani Devireddy, Mr. Tamer Elsherbeni, Ms. Fang Guo, Mr. Xiaolin Zhu, Mr. Xiao Liu, Mr. Jun Luo, Mr. Xiangfei Xu, Mr. Lin Gao, Ms. Xue Yang, Mr. Martye Hickman, Mr. Junling Zhao, and Mr. Lin Xiong. We also greatly appreciate the generous contributions of ANSYS Inc. in providing us with HFSS and Designer simulation software, and Rogers Corp. for providing us with high quality laminates that have been used for many of the designs that are presented in this book.

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1 Introduction to Reflectarray Antennas 1.1 ­Reflectarray Concept Communicating over long distances had long been a dream for mankind until 1901 when Marconi demonstrated the first cross Atlantic wireless signal transmission. Since then, long distance communications have evolved to a degree where mankind can communicate wirelessly across the Solar System and beyond. Long distance communication requires large antennas in order to establish the wireless link between the transmitter and receiver. One of the most practical types of electrically large antennas are reflectors. While reflectors were originally built as optical devices [1], the discovery of electromagnetic waves by Maxwell, began a new era for communication with these antennas. The first experimental demonstration of wireless communication by Hertz in 1887, used a dipole‐fed cylindrical parabolic antenna, which is believed to be the first reflector antenna operating at non‐optical frequencies. Since then, reflectors have become the most widely used high‐gain antenna in communications, radio astronomy, remote sensing, and radar [2]. An alternative approach to realization of a large antenna is by using several smaller antennas in the form of an array [3]. The first antenna array was built over 100 years ago [4]. In order to increase the directivity of a single monopole, Brown used two vertical antennas separated by half a wavelength and fed them out of phase [5]. He and several other notable scientists such as Marconi, Braun, and Adcock explored the unique characteristics of antenna array over the years [6]–[8]. Antenna array engineering evolved rapidly thereafter, particularly during the Second World War; however, it was the development of semiconductor technology in the 1960s and the printed circuit board technology in the 1970s that had the largest impact on their development. In particular the microstrip patch antenna proposed by Deschamps in 1953 [9] and later made practical by Munson in 1972 [10], revolutionized array engineering. Microstrip antenna arrays have since then played an important role in modern phased array systems. While reflectors and arrays still compete for large aperture jobs in many types of systems, in the recent years, a new generation of high‐gain antennas has emerged, which have attracted increasing interest from the antenna/electromagnetic community because of their low‐profile, low‐mass, and in many cases, low‐cost features. This antenna is known as the reflectarray antenna [11]–[13]. The reflectarray antenna is a hybrid design, which combines many favorable features of reflectors and printed arrays, and as Reflectarray Antennas: Theory, Designs, and Applications, First Edition. Payam Nayeri, Fan Yang, and Atef Z. Elsherbeni. © 2018 John Wiley & Sons Ltd. Published 2018 by John Wiley & Sons Ltd.

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Reflectarray Antennas

Figure 1.1  The geometry of an offset‐fed reflectarray antenna.

such can provide advantages over these two conventional antennas. The parabolic reflector is difficult to manufacture in many cases due to its curved surface that requires expensive custom molds and also become more difficult to manufacture at higher microwave frequencies. On the other hand, while antenna arrays offer the advantages of flexible design freedoms and versatile radiation performance, its feeding network suffers from the energy loss and design complexity, and the cost of the T/R modules [14] in active phased arrays becomes prohibitively high for many applications. As such, the reflectarray has fast been gaining attention as an alternative to these more mature technologies as it is able to mitigate the disadvantages associated with both of these high‐gain antennas. The reflectarray is an antenna with a flat reflecting surface consisting of hundreds of elements on its aperture and an illuminating feed antenna, as shown in Figure 1.1. The feed antenna spatially illuminates the aperture where the elements are designed to reflect the incident field with certain phase shifts in order to collimate the beam of the antenna in the desired direction and with the preferred shape. Its operation principle is similar in concept to reflector antennas with respect to the spatial illumination, and again similar in concept to antenna arrays with respect to phase synthesis and beam collimation.

1.2 ­Reflectarray Developments The concept of reflectarray antennas was initially introduced in the early 1960s using short‐ended waveguide elements with variable lengths [11]. The feed antenna illuminated the waveguides where the lengths of the shorted waveguides were designed such that the phase of the reradiated signals would form a collimated beam in the desired far‐field direction. While the concept was very interesting, the bulky and heavy

Introduction to Reflectarray Antennas

Figure 1.2  The first reflectarray antenna using waveguide technology. Source: Berry 1963 [11]. Reproduced with permission from IEEE.

waveguide structure of this first reflectarray antenna was a major drawback. The experimental model of the waveguide reflectarray is shown in Figure 1.2. Although some work on spiralphase reflectarrays was reported by Phelan in the mid‐1970s [15], the reflectarray antenna did not receive much attention after that until the revolutionary breakthrough of printed microstrip antenna technology in the 1980s. Since then, research on reflectarray antennas has been on the rise, and several diversified applications such as multi‐beam antennas for point‐to‐point communication, beam‐scanning antennas for radar applications, and spatial power combining reflectarray systems have been demonstrated. In particular, over the past 10 years, an increased interest in reflectarray antenna research has been observed in both academic and industrial sectors of the antenna community, which is also propelled by advances in fabrication technologies as well as computational resources. Since 2006, the IEEE Antennas and Propagation International Symposium (APS) has included sessions dedicated to reflectarray antennas in the general conference proceedings, and several sessions and special sessions have been held since then. Most notably a full‐day special session on reflectarray antennas was held at the 2011 APS. Several hundred papers have been presented in these sessions, and many researchers are now interested in joining this active research area. In 2012, the International Journal of Antennas and Propagation published a special issue on Reflectarray Antennas: Analysis and Synthesis Techniques, which further stimulated the research interest in this area. A literature search on IEEE Xplore using the keyword “reflectarray” showed more than 1200 articles have been published in IEEE in this area, as shown in Figure 1.3. The majority of the articles, however, have been published in the recent years, and in particular, there has a notable increase in the number of papers over the last 10 years. The reflectarray antenna offers a multitude of capabilities that has encouraged continuous development and exciting applications in recent years. The elements of the reflectarray are designed to reflect the electromagnetic wave with a certain phase to compensate for the phase delay caused by the spatial feed. The phase shift of the elements is realized using various methods such as variable size elements, phase‐delay lines, and element rotation techniques. The infinite array approach is used to calibrate the element phase versus element change [12]. Due to the very large number of elements

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Reflectarray Antennas 200

150 Number of Articles

4

100

50

0 1990

1995

2000

2005

2010

2015

Year

Figure 1.3  The number of articles on reflectarray antennas published in IEEE. Data obtained from IEEE Xplore on April 1, 2016.

involved in a reflectarray, full‐wave simulation of the entire reflectarray antenna is still challenging. On the other hand, different theoretical models have been developed for the analysis of reflectarrays, such as the array theory formulation and the aperture field analysis technique, which show a good agreement with measured results. Moreover, implementing the spectral transform in these calculations allows for fast calculation of the radiation characteristics of the antenna, which is a considerable advantage for synthesis design problems using iterative procedures. Single and multilayer reflectarrays have been designed to achieve broadband and multiband performance from microwave frequencies up to the THz range [16], [17]. Considerable improvements have been made to these designs over the years, and many practical designs have been demonstrated. One of the main challenges in reflectarray designs is improving the bandwidth of the antenna, which is the major drawback of printed resonator‐type structures [18]. Different bandwidth improvement techniques such as using multilayer designs [19], [20], true time‐delay lines [21], and sub‐wavelength elements [22] have been studied and bandwidths of more than 20% have been reported. Meanwhile, the direct control of the phase of every element in the array allows multi‐beam performance with single or multiple feeds. The design of contoured beam reflectarrays is also a challenging field [23]. A phase‐only synthesis process is used to obtain the required element phase shift from any given mask. Multi‐feed multi‐beam contoured beam designs have been demonstrated [24]; however, the performances of these designs are slightly inferior to the shaped‐beam parabolic reflectors. Another advantage of reflectarrays is the ability of the antenna to scan the main beam to large angles off broadside. Beam‐scanning reflectarrays are designed by using low‐loss phase shifters integrated in every element of the array [25]. These beam‐scanning reflectarrays require a switch board to control the main beam direction and are well suited for radar applications, and some models have been demonstrated; however, considerable challenges lie in improving the performance of these beam‐scanning antennas.

Introduction to Reflectarray Antennas

In addition to the numerous capabilities and potentials that reflectarray antennas have demonstrated, a great deal of interest is now in the practical implementation of reflectarray antennas for space applications. Since the common considerations for space antennas are size, weight, and power (SWaP), because of limitations imposed by the satellite launch capabilities [26], the reflectarray antenna shows significant advantages over conventional high‐gain space antennas, which are typically reflectors/ lenses and arrays. These momentous promises make the reflectarray antenna a suitable low‐cost choice for the new generation of space antennas. The advantages of reflectarrays, such as low‐profile, lightweight, and conformal geometry, make it desirable for various communication systems, especially for those mobile platforms. Its applications in space exploration, satellite communications, remote sensing, and radar systems are rising up within the last decade, and will continue to increase in the future. In addition, the current printed circuit board (PCB) fabrication technology and available low‐cost commercial laminates, allows for low‐cost rapid prototype fabrication. This is also leading to commercial implementation and large‐scale fabrication of reflectarray antennas for commercial applications. Terahertz and optical applications are also a very promising future of reflectarrays. With advances in fabrication technologies such as 3D printing devices and nanotechnology, the practical implementation of THz and optical reflectarray designs at a competitive cost is not far away. The full potential of reflectarray capabilities has not yet been fully exploited. Researchers in this field are constantly presenting new ideas and designs ranging from advanced materials to multifunctional system designs. As such it is expected that this field will remain an active area of research in the next decade, and there is no doubt that reflectarrays will become an important member in the antenna family.

1.3 ­Overview of this Book The aim of the book is to provide scientists and engineers in the fields of antenna, microwave, and electromagnetic, with an up‐to‐date knowledge of reflectarray antenna theories as well as the design and analysis techniques. This book will provide the reader with: ●●

●●

●●

An overview of the reflectarray antenna research history, including various implementations and state‐of‐the‐art. A good knowledge of the basic theories for design and analysis of reflectarray antennas, which will help to build up the fundamental capabilities for reflectarray research. In addition, a wealth of design examples along with numerical and experimental results are presented, which serves as a reference for researchers to verify their own developed programs. Detailed design procedures for a wide range of diversified applications, such as broadband designs, multiband operation, multi‐beam performance, contour‐beams, beam‐scanning systems, and conformal reflectarray antennas, along with illustrative examples for each design.

The prerequisite for this book is that the readers should be familiar with the basics of antenna engineering. An introductory course to antenna engineering is typically offered as a senior level course for a bachelor student in the field of electrical engineering.

5

6

Reflectarray Antennas Part I: Reflectarray Basics Chapter 2 Element Design

Chapter 3 System Analysis

Chapter 4 Pattern Calculation

Chapter 5 Bandwidth Discussion

Chapter 6: Design Examples

Part II: Advanced Reflectarray Designs Chapter 7 Broadband / Multiband Reflectarrays

Chapter 8 THz / Infrared / Optic Reflectarrays

Chapter 9 Multi-beam / Shaped-beam Reflectarrays

Chapter 10 Beam Scanning Reflectarrays

Chapter 11 Advanced configurations and Engineering Applications

Figure 1.4  Organization of this reflectarray book.

As such any student in this field will be able to benefit from this book. However, this book is intended for both beginners and specialists in the field of electrical engineering. This is achieved by organizing and preparing this book in two parts and in 11 chapters, as illustrated in Figure 1.4. The first part, which includes the fundamental theories of reflectarrays, is intended for engineers that know the basics of antenna theory and are starting to become familiar with this new generation of high‐gain antennas. The second part of the book is intended for researchers and specialists that have a good knowledge of the basic theories in reflectarray, and aim at designing reflectarray antennas for specific applications/operations. The first part includes the basic theories for analysis and design of reflectarray antennas. This section of the book builds the fundamental knowledge one needs to have in order to understand the governing dynamics of a reflectarray antenna system, and efficiently design and analyze reflectarray antennas. Chapter 2 is devoted to analysis and design of reflectarray phasing elements, and provides a comprehensive coverage of aperture phase requirements in reflectarray systems, phasing element design methodologies, element analysis techniques, as well as design examples. The reflectarray system design is introduced in Chapter 3, where the readers will learn the basics of the reflectarray systems and efficiency analysis for practical designs. A detailed coverage of the various methods to compute the radiation characteristics of reflectarray antennas is presented in Chapter 4. The bandwidth characteristics of reflectarray antennas is studied in detail in Chapter 5. The last chapter of the first part of this book is devoted to design examples, where a variety of reflectarray designs are presented that can serve as a useful reference for interested readers.

Introduction to Reflectarray Antennas

The second part of the book is intended for researchers that have a good knowledge of the basic theories in reflectarray, and aim at designing reflectarray antennas for specific applications/operations. This part starts with a comprehensive chapter on broadband and multiband reflectarray antennas in Chapter 7. Reflectarrays operating above microwave frequencies such as in the terahertz, infrared, and optical spectrums are introduced in Chapter 8. After discussion of the frequency behaviors of reflectarrays, advanced designs on the radiation patterns are followed. A detailed coverage of multi‐beam and shaped‐beam reflectarrays is presented in Chapter  9. Chapter 10 is devoted to beam‐scanning reflectarray antennas, where the extensive research on these types of reflectarrays are summarized and analyzed in a comprehensive fashion. The final chapter of this book is devoted to advanced configurations of reflectarray antennas, such as conformal geometries and dual‐reflector configurations, and applications such as satellite communications and spatial power combining.

­References 1 A. Dodd, “An acre of glass  –  The history of the telescope,” [Online]. Available: http:/

2 3 4 5 6

7 8 9 10 11 12 13 14 15

ezinearticles.com/?An‐Acre‐of‐Glass–The‐History‐of‐the‐Telescope&id=2601009 (accessed July 1, 2017). Y. Rahmat‐Samii and R. L. Haupt, “Reflector antenna developments: A perspective on the past, present and future,” IEEE AP‐S Mag, Vol. 57, No. 2, Apr 2015. R. L. Haupt and Y. Rahmat‐Samii, “Antenna array developments: A perspective on the past, present and future,” IEEE AP‐S Mag, Vol. 57, No. 1, Feb 2015. J. A. Fleming, The Principles of Electric Wave Telegraphy and Telephony, 3rd Edn, New York: Longmans, Green, and Co., 1916. S. G. Brown, Brit. Patent No. 14,449, 1899. G. Marconi, “On methods whereby the radiation of electric waves may be mainly confined to certain directions, and whereby the receptivity of a receiver may be restricted to electric waves emanating from certain directions,” Proc. Roy. Soc. Lond., Ser. A., Vol. 77, p. 413, 1906. F. Braun, “Electrical oscillations and wireless telegraphy,” Nobel Lecture, Dec. 11, 1909. F. Adcock,Improvement in means for determining the direction of a distant source of electro‐magnetic radiation, UK Patent 130,490, Aug. 7, 1919. G. A. Deschamps, “Microstrip microwave antennas,” presented at the 3rd USAF Symp. on Antennas, 1953. R. E. Munson, “Microstrip phased array antennas,” Proc. of 22nd Symp. on USAF Antenna Research and Development Program, Oct 1972. D. G. Berry, R. G. Malech, and W. A. Kennedy, “The reflectarray antenna,” IEEE Trans. Antennas Propagat., Vol. AP ‐ 11, Nov. 1963, pp. 645–651. D. M. Pozar, S. D. Targonski, and H. D. Syrigos, “Design of millimeter wave microstrip reflectarrays,” Proc. IEEE Trans. Antennas Propagat., Vol. 45, pp. 287–295, Feb. 1997. J. Huang and J. A. Encinar, Reflectarray Antennas. New York, NY, USA: Wiley‐IEEE, 2008. R. C. Hansen, Phased Array Antennas, 2nd Edn, Chichester, UK: John Wiley & Sons, Ltd, 2009. H. R. Phelan, “Spiralphase reflectarray for multitarget radar,” Microwave Journal, Vol. 20, July 1977, pp. 67–73.

7

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Reflectarray Antennas

16 A. Yu, “Microstrip reflectarray antennas: Modeling, design and measurement,” Ph.D.

dissertation, Dept. Elect. Eng., University of Mississippi, Oxford, MS, 2010.

17 P. Nayeri “Advanced design methodologies and novel applications of reflectarray

18 19 20

21

22

23 24

25

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antennas,” Ph.D. dissertation, Department of Electrical Engineering, University of Mississippi, MS, 2012. D. M. Pozar, “Bandwidth of reflectarrays”, Electronics Letters, Vol. 39, No. 21, Oct. 2003. J. A. Encinar, “Design of two‐layer printed reflectarrays using patches of variable size”, IEEE Trans. Antennas Propag., Vol. 49, No. 10, pp. 1403–1410, Oct. 2001. J. A. Encinar and J. A. Zornoza, “Three‐layer printed reflectarrays for contoured beam space applications,” IEEE Trans. Antennas Propag., Vol. 52, No. 5, pp. 1138–1148, May 2004. E. Carrasco, J. A. Encinar, and M. Barba, “Bandwidth improvement in large reflectarrays by using true‐time delay”, IEEE Trans. Antennas Propag., Vol. 56, No. 8, pp. 2496–2503, Aug. 2008. P. Nayeri, F. Yang, and A. Z. Elsherbeni, “A broadband reflectarray using sub‐wavelength patch elements,” IEEE Antennas and Propagation Society International Symposium, South Carolina, U.S., 2009. D. M. Pozar, S. D. Targonski, and R. Pokuls, “A shaped‐beam microstrip patch reflectarray,” IEEE Trans. Antennas Propag., Vol. 47, pp. 1167–1173, July 1999. M. Arrebola, J. A. Encinar, and M. Barba, “Multifed printed reflectarray with three simultaneous shaped beams for LMDS central station antenna”, IEEE Trans. Antennas Propag., Vol. 56, No. 6, pp. 1518–1527, June 2008. S. V. Hum, M. Okoniewski, and R. J. Davies, “Modeling and design of electronically tunable reflectarrays,” IEEE Trans. Antennas Propagat., Vol. 55, No. 8, pp. 2200–2210, Aug. 2007. R. B. Dybdal, “Satellite antennas,” in Antenna Engineering Handbook, J. Volakis (ed.), McGraw‐Hill, 2007.

9

2 Analysis and Design of Reflectarray Elements A reflectarray antenna consists of a planar or conformal array of elements that are excited with a feed antenna [1]–[3]. A typical model of a reflectarray antenna is given in Figure 2.1. Each element is designed such that when it is illuminated by the feed antenna, it incorporates a certain reflected phase. The phase distribution over the reflectarray aperture is then synthesized so the reflectarray can realize a collimated or shaped beam in the desired direction. As such, analysis and design of the reflectarray elements, typically referred to as phasing elements, is of paramount importance. There are two steps in the design of a reflectarray, namely, the element design and the system design. The element design will be discussed in this chapter, and the system design will be discussed in the following chapters. In this chapter, we will first study the basics of designing the phase distribution on the reflectarray aperture. Next, we will outline the phase tuning approaches for reflectarray elements. In other words, how the individual elements are designed to scatter electromagnetic waves with the desired phases. Moreover, numerical and analytical approaches for analysis of reflectarray phasing elements will be outlined, and several examples of reflectarray phasing elements will be presented. Some discussions on frequency behavior, effects of oblique excitation, and sources of phase error for reflectarray elements will also be presented.

2.1 ­Phase‐Shift Distribution on the Reflectarray Aperture In classic planar antenna arrays, a uniform phase distribution on the aperture will yield a collimated beam in broadside direction, that is, normal to the plane of the array. To focus the beam in a certain direction, a progressive phase distribution is assigned to the elements [4], [5]. For reflectarrays, the basic operating principle is similar, however, one also needs to account for the feed antenna position [6]. The feed antenna is located at a certain position with respect to the reflectarray coordinate system, as shown in Figure 2.1. Typically, the elements of the reflectarray are assumed to be in the far field of the feed antenna; therefore, the incident electromagnetic field on each reflectarray element can be approximated by a plane wave that excites the element with a certain incident angle. The electromagnetic fields emanating from the feed, propagate as a spherical wave which originate from the phase center of the feed antenna. The incident electromagnetic fields on the reflectarray aperture have a phase proportional Reflectarray Antennas: Theory, Designs, and Applications, First Edition. Payam Nayeri, Fan Yang, and Atef Z. Elsherbeni. © 2018 John Wiley & Sons Ltd. Published 2018 by John Wiley & Sons Ltd.

10

Reflectarray Antennas Z

Y X

Figure 2.1  Typical geometry of a planar reflectarray antenna.

Figure 2.2  Typical geometrical parameters of a planar reflectarray antenna.

Z

Phase Center of Feed Antenna

rˆo θo

Ri

Y

φo ri ith element

X

to the distance they traveled, which is referred to as spatial phase delay. As such, in order to achieve a collimated beam, the phasing elements of the reflectarray have to compensate for this phase. A geometrical model of the reflectarray system, showing the position of the feed phase center, and the reflectarray coordinate system is given in Figure 2.2. The reflection phase of a reflectarray element should compensate for the spatial phase delay (spd) from the feed phase center to that element. Mathematically this is given by

spd

k0 Ri , (2.1)

where Ri is the distance from the feed phase center to the ith element, and k0 is the wavenumber at the center frequency. Such a phase distribution converts the spherical wave radiated by the feed antenna, to a collimated beam in the broadside direction, that

Analysis and Design of Reflectarray Elements

is, in the Z‐direction with respect to Figure 2.2. To scan this collimated beam in any direction, a progressive phase (pp) can be added to the aperture, which in vector form is given by  φ pp = −k0 ri .rˆo , (2.2)  where ri is the position vector of the ith element, and rˆo is the direction of the main beam as shown in Figure 2.2. In the Cartesian coordinate system of Figure 2.2, the position of each element can be expressed as (xi, yi), thus for a beam directed in a certain spherical direction (θo, φo), this equation simplifies to

pp

k0 xi sin

o cos o

yi sin

o sin

o

. (2.3)

The required phase shift on the reflectarray aperture therefore needs to compensate for the spatial delay (‐spd) and add the progressive phase to the aperture, which is given by

RA

k0 Ri sin

o

xi cos

o

yi sin

o

0 , (2.4)

where ϕ0 is a phase constant, indicating that a relative phase is needed for the reflectarray elements. It should be noted that the required phase distribution given in (2.4), will produce a pencil beam in the desired direction. For shaped beams, however, the progressive phase in (2.3) should be replaced by an appropriate phase distribution, which is obtained using a phase‐only synthesis and will be discussed in later chapters. In all designs, however, the phasing elements of the reflectarray antenna must compensate for the spatial phase delay associated with the feed. The reflectarray antenna is basically analogous to a parabolic reflector antenna, however, unlike the parabola, the reflectarray consists of a certain number of elements, typically arranged in a Cartesian grid. As such, the phase distribution on the reflectarray aperture is essentially pixelated. As an example, the grid layout of a Ka‐band reflectarray antenna with a circular aperture is given in Figure 2.3, where each element is assumed to have a constant phase over its own aperture. Figure 2.3  Grid layout of the unit‐cells of a reflectarray antenna with circular aperture.

11

Reflectarray Antennas 40 300 30 200

20

100

10

10

20

30

40

y-axis [element number]

y-axis [element number]

40

300 30 200

20

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10

0

10

x-axis [element number]

20

30

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0

x-axis [element number]

(a)

(b) 40 300

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0

100

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150

0

x-axis (mm)

x-axis [element number]

(c)

300

150

30 y-axis (mm)

y-axis [element number]

12

(d)

Figure 2.4  Phases on the aperture with a centered feed and broadside beam: (a) spatial delay, (b) progressive phase, (c) phase distribution on the reflectarray antenna, and (d) phase distribution on the continuous aperture.

The circular aperture of the reflectarray has a diameter of 190 mm, which is about 20 wavelengths at 32 GHz. The element spacing is 4.7 mm, that is, each element of the array has a lattice size of 4.7 × 4.7 mm2. In total, this array has 1184 elements. Figure  2.4(a) shows the spatial phase delay on the aperture when the feed phase center is placed along the z‐axis at a distance of 170 mm from the aperture, see Figure 2.2. The array is designed to have a beam in the broadside direction, hence no progressive phase is required as shown in Figure 2.4(b). The phase distribution on the reflectarray aperture along with the continuous phase distribution are also given in Figure 2.4(c) and (d), respectively. It can be seen that the phase moves outwards from the center of the array, and almost three complete phase cycles (0 to 2π) is observed with this design. Similar to reflectors, the centered feed configuration exhibits a number of disadvantages such as the feed blockage, which will be outlined in a later chapter. As such, typically an offset configuration is desirable. With the same aperture parameters as the previous example, the phase distributions of an offset configuration with an off‐broadside beam given in Figure 2.5. The feed antenna phase center is placed in the xz‐plane, and has an

Analysis and Design of Reflectarray Elements 40 300 30 200 20 100

10

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40

y-axis [element number]

y-axis [element number]

40

300 30 200

20

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0

10

x-axis [element number]

20

30

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40

x-axis [element number]

(a)

(b)

300 30 200

20

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0

50

x-axis [element number]

(c)

300

150 y-axis (mm)

y-axis [element number]

40

100

150

0

x-axis (mm)

(d)

Figure 2.5  Phases on the aperture with an offset feed and an off‐broadside beam: (a) spatial delay, (b) progressive phase, (c) phase distribution on the reflectarray antenna, and (d) phase distribution on the continuous aperture.

offset angle of 30°, that is, x = −85 mm and z = 147.22 mm with respect to the coordinate system of Figure 2.2. The feed points to the geometrical center of the aperture, and the array is designed to focus the beam at (θo = 30°, φo = 0°). The phase distributions presented here clearly show that the elements of a reflectarray must be able to provide a complete phase tuning range. In other words, a mechanism is needed to change the reflection phase of the elements from 0 to 2π. In Section 2.2, we will outline three different phase tuning techniques that have been developed to address this fundamental requirement.

2.2 ­Phase Tuning Approaches for Reflectarray Elements A critical step in designing a reflectarray antenna, is selecting a phase tuning methodology that enables the reflectarray elements to achieve the desired phase tuning range. Once the phase tuning approach is selected, the element characteristics can be determined, which are ultimately used to realize the radiation characteristics of the antenna.

13

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Reflectarray Antennas

Different approaches are available for tuning the phase of reflectarray elements, which in general can be categorized into three groups: 1) Elements with phase/time‐delay lines, 2) Elements with variable sizes, 3) Elements with variable rotation angles. In the first group, the incident wave is first received by the element, then phase shifted using a delay‐line microwave network, and finally re‐radiated. This methodology is known as the guided‐wave technique, and is essentially the concept that Berry [1] used when introducing the first reflectarray antenna in 1963 that utilized short‐circuited waveguide elements. With the second group, variable size elements, the phase of the scattered field is controlled by changing the dimensions of the element, for example, the patch length. The last group, which is applicable only to circularly polarized (CP) designs, takes advantage of the unique property of an incoming CP wave, that is, the rotational wave vectors, to achieve phase tuning. In this section, we will outline the basic principles of these three phase tuning approaches. 2.2.1  Elements with Phase/Time‐Delay Lines In this configuration, the element (typically a patch) receives the electromagnetic wave from the feed antenna, and transfers it into a guided wave along a transmission line (typically a microstrip line) with a certain length [7]–[9]. The transmission line can be terminated in either open or short circuit. The signal reflects back from the transmission line termination, and is re‐radiated by the element. A schematic model of this type of element is given in Figure 2.6. In this approach, the phase shift is proportional to twice the line length, which is given by

delay line

2kl , (2.5)

where l is the length of the line, and k is the propagation constant of the signal along the transmission line. The differential phase shift between any two elements of the reflectarray, Δϕ, can then be achieved by setting the physical length difference, Δl, equal to Δϕ/2k. In the ideal case, the reflection phase of these types of elements will be perfectly linear, however, the specular reflection from the ground plane, as well as the stub

Figure 2.6  Schematic models of reflectarray patch elements with attached phase/time‐delay lines of two different lengths.

Analysis and Design of Reflectarray Elements

resonances which occur for certain lengths, and the dissipative losses of the bends in the transmission line, will distort the phase response. The design principle for this approach is similar to classic microstrip antennas, where the patch dimensions must be suitably chosen such that it resonates at the frequency of incident wave. The line (stub) should also be matched to the patch to allow for proper guided‐wave propagation. This is particularly important, since if there is mismatch between the patch and the line, part of the signal will be reflected by the patch before going through the line. In this case, the reflected wave will be the sum of two components, and the reflection phase will no longer be proportional to twice the length of the line. It is also critical to note here that to account for mutual coupling between the elements of the reflectarray antenna, design of the patch and delay line should be carried out when the element is placed in the array environment, not as an isolated element. Detailed discussions on analysis techniques for reflectarray elements will be given in Section 2.3. 2.2.2  Elements with Variable Sizes In this approach, the physical size of the element is changed to provide phase tuning. Theoretically, it is known that changing the length of a resonant element changes the resonance frequency of the antenna, which corresponds to a change in radiated phase at a certain frequency. The operating principle of the variable size technique is therefore based on the fact that the reflected phase from resonant elements with different sizes would be different. This unique method was first introduced in [10] for printed crossed dipoles, and in [11] for rectangular patches. The variable element size technique, although conventionally adopts square or circular patch geometry, provides the most diversified choices of element geometries [12]. A printed antenna is a highly resonant element with a large quality factor, therefore a small change in its size produces a wide range of phase change in the reflected wave. Ideally, a single resonance can provide a complete phase cycle of 360°, however, in practice, the total achievable phase range depends on a number of factors such as the separation between patches and the substrate thickness, and is less than a complete cycle. For small substrate thickness, typically less than one tenth of the wavelength, a phase tuning range above 300° can usually be achieved, which is sufficient for most reflectarray designs. A schematic model of this type of element is given in Figure 2.7. It is important to note that the phase variation obtained using this approach is highly non‐linear primarily because of the high‐Q resonant nature of printed antennas with thin substrates. This results in a rapid phase variation near resonance, and slow Figure 2.7  Schematic models of variable size reflectarray square patch elements of two different sizes.

15

Reflectarray Antennas

150

Phase Change (degrees)

16

100 50 0 –50 –100 –150 –Δ

0 Element Change

+Δ

Figure 2.8  A typical S‐curve for variable size reflectarray elements.

variations at the extreme dimensions. While the shape of the curve generally depends on the element design scheme, typically it is in the form of an S‐shaped curve. A typical S‐curve for reflectarray elements is given in Figure 2.8. 2.2.3  Elements with Variable Rotation Angles A very clever phase tuning technique, which is restricted to CP designs, is the element rotation approach. This technique is based on the fact that rotating a CP antenna element about its origin by ψ°, will change the radiated phase by the same amount where the advance or delay of the phase depends on the direction of rotation [13]. This technique was first applied to reflectarrays using patch elements with attached phase‐delay lines [14]. The patches were rotated about their origin (center of the unit‐cell), resulting in a phase shift for the reflected wave, which is a linear function of the rotation angle. To utilize this technique for phase tuning of reflectarray elements, one needs to determine the direct relationship between rotation angle and phase of the reflected wave. The mathematical formulation is outlined here. Without the loss of generality, we consider a right‐hand circularly polarized (RHCP) wave traveling in the –z‐direction impinging on the unit‐cell, as schematically shown in Figure 2.9. Mathematically this is given by  Ei = E0 ( xˆ + jyˆ ) e jk0 z e jωt , (2.6) where k is the wavenumber and is the angular frequency of the wave. The reflected wave propagates in the + z‐direction, therefore in the usual manner, the reflected wave will be converted to a left‐hand circularly polarized (LHCP) wave, due to the reversal of the direction of propagation. This phenomenon is observed in reflectarrays with dual‐linear elements (such as square patches) and also in reflector antennas. In the case of reflectarrays, however, if the element can introduce a 180° phase difference between

Analysis and Design of Reflectarray Elements

Figure 2.9  A 3D schematic model of a CP reflectarray element.

Plane Wave –z

z x

x

y

x x′ ψ

y

y

y′

Figure 2.10  A schematic model of a CP reflectarray element showing the reference element with 0° phase shift (left) and the ψ rotated element with 2ψ phase shift (right).

the two orthogonal components of the reflected field, the sense of polarization can be maintained. In this setup, this requires the phase in the y‐direction (ϕy) to lead by 180°, that is, y x 180 . This can be realized with phase‐delay lines, or elements that can intrinsically create this phase difference, such as split rings. For the reference element in Figure 2.10(a), the reflected field can be written as  ˆ jφx + jye ˆ jφy e − jk0 z e jωt = E0 ( xˆ − jyˆ ) e jφx e − jk0 z e jωt , (2.7) Er = E0 xe

(

)

where it is assumed that there is no loss due to reflection from the element. Now let us consider the case when the element is rotated by ψ, as shown in Figure 2.10(b). The incident and reflected fields can now be easily obtained in the new coordinate system (x′,y′), which is aligned with the rotated element. The incident field is given by  Ei = E0 ( ( xˆ′ cosψ − yˆ′ sinψ ) + j ( xˆ′ sinψ + yˆ′ cosψ ) ) e jk0 z e jωt , (2.8) which after some algebraic manipulation can be written as  Ei = E0 ( xˆ′ + jyˆ′ ) e jψ e jk0 z e jωt . (2.9) The reflected field for the rotated element with

y

x

180 is now given by

 Er = E0 xˆ′e jφx′ + jyˆ′e jφy′ e jψ e − jk0 z e jωt = E0 ( xˆ′ − jyˆ′ ) e jφx′ e jψ e − jk0 z e jωt . (2.10)

(

)

17

18

Reflectarray Antennas

Note that since the element is symmetric with rotation, such as the circular split ring in Figure 2.10, x x , y y , and the reflected field can then be written as

 Er = E0 ( xˆ′ − jyˆ′ ) e jφx e jψ e − jk0 z e jωt . (2.11)

Re‐writing the reflected fields in terms of the original (x, y) coordinates yields  Er = E0 ( xˆ ( cosψ + j sinψ ) + yˆ ( sinψ − j cosψ ) ) e jφx e jψ e − jk0 z e jωt . (2.12) With some algebraic manipulation, the reflected field can be written as

 Er = E0 ( xˆ − jyˆ ) e jφx e j 2ψ e − jk0 z e jωt . (2.13)

Comparing (2.13) to (2.7), the reflected field is delayed in phase by 2ψ when the element is rotated counter clockwise by ψ about its origin. Similarly, it can be shown that a LHCP wave would be advanced in phase when the element is rotated counter clockwise. It is also important to point out that in the analysis presented here, the cross‐polarized reflected fields were assumed to be zero. This can be taken into account by expanding (2.7) with proper reflection magnitudes and phases, however, note that the phase of the cross‐polarized component is different from the phase of the co‐polarized component discussed previously.

2.3 ­Element Analysis Methods In general, for analysis and design of the elements of an antenna array, one needs to take into account the electromagnetic interaction between the elements, which is known as mutual coupling. The effect of mutual coupling is serious and needs to be accounted for, if the element spacing is small compared to the wavelength, which is usually the case for arrays. On the other hand, for large antenna arrays with several hundreds of elements, such as reflectarrays, it is very challenging to conduct the analysis in the full array model. In an array with a very large number of regularly spaced elements, all the elements except those near the edge have approximately the same behavior. In an infinite array [2], every element has identical behavior because there is no edge or other variation in the arrangement of the elements. The basic properties of nearly all the elements in a very large regular array can therefore be represented by an element in the corresponding infinite array, which has firmly been established as the best basis for the element design in large finite arrays. It should be noted here that the infinite array approach assumes all surrounding elements are identical, in other words the structure that is analyzed is assumed to be periodic. In the case of reflectarrays, however, the elements are not identical, and the array is quasi‐periodic in nature. Nonetheless, in most cases, this periodic approximation is adequate enough to characterize the properties of the reflectarray elements. In this section, we will study different techniques that have been developed over the years for efficient analysis of reflectarray elements.

Analysis and Design of Reflectarray Elements

2.3.1  Periodic Boundary Conditions and Floquet Port Excitation One technique that can mimic a plane wave impinging normally to the element aperture is the H‐wall waveguide simulator, also known as the parallel waveguide simulator [15]–[17]. In most cases, however, it is desirable to study the reflection characteristics of the element, when it is excited with a plane wave that impinges in a general direction (θi, φi) and polarization (TE, TM). The wave then reflects back in the specular direction with a set of amplitude and phase distributions. Although such an H‐wall waveguide simulator can accurately mimic a normal plane wave excitation, a more generalized approach will be outlined here that can simulate both normal and oblique excitations. In this approach, the element is placed at the end of a rectangular cuboid where the width and length of the cross‐section is the same as the unit‐cell size [17]. A geometrical model of this element excitation setup is given in Figure 2.11. The four sidewalls of the unit‐cell are linked together as periodic boundaries, which enables one to model planes of periodicity where the field on one surface matches the field on another with a certain phase difference. The unit‐cell is then excited with a Floquet port, which is located on the top surface of the cuboid, typically at a distance of λ/2 at the minimum frequency. The Floquet port excites a set of Floquet modes, which are fundamentally plane waves with propagation direction set by the frequency and geometry of the periodic structure. This is the general setup for analysis of a periodic array element for any angle of incidence or sense of polarization. The reflected field quantities can then obtained directly at the port. It is important to note that several commercial electromagnetic simulation software are available for this analysis such as Ansys HFSS [18], CST Microwave Studio [19], and FEKO [20]. 2.3.2  Metallic Waveguide Simulators The metallic waveguide simulator can accurately characterize the reflection properties of the element under certain conditions. In this technique, the element is placed at the end of a real waveguide and is excited with the dominant TE10 mode. This waveguide setup essentially creates an infinite array scenario for the element, which can be explained using image theory [16]. The elements in the waveguide will be excited with an oblique angle given by



cos

1

1

fc f

2

cos

1

1

2

2a

. (2.14)

Here fc is the cutoff frequency of the fundamental mode, which is directly related to a, the larger dimension of the rectangular waveguide. Each waveguide can only simulate one incident angle at one frequency, therefore, several different waveguides need to be constructed in order to analyze the element with several different excitation angles at multiple frequencies. Due to these limitations, this approach is seldom used for the element simulation. However, it is particularly useful for measuring the element performance.

19

20

Reflectarray Antennas

(a) E Field [v_per_... 1.0000e + 004 9.3571e + 003 8.7143e + 003 8.0714e + 003 7.4286e + 003 6.7857e + 003 6.1429e + 003 5.5000e + 003 4.8571e + 003 4.2143e + 003 3.5714e + 003 2.9286e + 003 2.2857e + 003 1.6429e + 003 1.0000e + 003

(b) Figure 2.11  A unit‐cell analysis setup: (a) 3D model in Ansys HFSS, (b) vector electric fields of the Floquet port excitation.

The standard rectangular waveguides typically have an aspect ratio about 2:1. By placing two elements in the waveguide, a unit‐cell with an almost square lattice can be simulated. A simulation setup for a square patch element inside a standard rectangular waveguide is shown in Figure 2.12(a). The vector electric fields at the waveguide port are also shown in Figure 2.12(b). Note that the waveguide excitation is not a uniform plane wave, however, it will allow one to accurately characterize the reflection properties of the reflectarray element. More discussion on this issue, and comparison with other approaches will be given in Section 2.3.4.

Analysis and Design of Reflectarray Elements

(a) E Field [v_per_... 2. 0000e + 003 1. 8572e + 003 1. 7143e + 003 1. 5715e + 003 1. 4286e + 003 1. 2858e + 003 1. 1429e + 003 1. 0001e + 003 8. 5726e + 002 7. 1441e + 002 5. 7157e + 002 4. 2873e + 002 2. 8589e + 002 1. 4304e + 002 2. 0000e – 001

(b) Figure 2.12  A metallic waveguide simulator: (a) 3D model in Ansys HFSS and (b) vector electric fields of the wave port excitation.

2.3.3  Analytical Circuit Models The unit‐cell can also be analyzed by viewing the structure as a transmission line circuit [21]. Depending on the geometrical features of the element, an analogous circuit model has to be derived for the analysis. For the square patch studied in this section, the metallic

21

22

Reflectarray Antennas

patches act as parallel capacitors with the adjacent cells while the short‐circuited unit‐cell ground can be modeled as a parallel inductor. The loss in the dielectric substrate can be modeled by a parallel resistor, that is, the total unit‐cell is modeled as a parallel RLC circuit. The conductor losses can also be represented by a resistor, which is connected in series with the capacitor as shown in [22]. Using the circuit model representation of the element, the impedance at the top surface of the element (Zs) can be calculated directly. The reflection coefficient Γ of an incident plane wave on the unit‐cell can then be calculated using transmission line theory, that is,

Zs Zs

Zo . (2.15) Zo

where Zo is the free space impedance and Zs is the surface impedance of the element obtained by the circuit model as shown in Figure 2.13. It is implicit that the main challenge in this analysis is deriving a correct analogous circuit model representation of the element, along with proper values for the lumped elements. Alternatively, once the circuit model has been obtained, it is possible to obtain accurate values for the lumped elements by designing a resonant circuit at the operating frequency and then tuning these values so that the circuit model reflection curves fit the curves obtained by the full‐wave unit‐cell simulations. It should also be pointed out that the circuit model analysis is general and the transmission line parameters can be derived for oblique incidence angles as well. The advantage of the circuit models is that they provide a simple and direct physical insight on the operation mechanism of the elements. 2.3.4  Comparison of Element Analysis Techniques In order to observe the accuracy of these element analysis methods, the three different techniques described in this section are compared together. As discussed earlier, the PBC with Floquet port excitation, and the circuit model techniques allow one to simulate the reflectarray element for any excitation angle, and with flexibility in the unit‐cell dimension. The metallic waveguide simulator, however, is more limited, since the dimensions of the unit‐cell are typically restricted to the waveguide dimensions, and the Γ

C

R L Zs

Figure 2.13  The circuit model for the unit‐cell of a reflectarray antenna.

Analysis and Design of Reflectarray Elements

waveguide can only simulate one incident angle at one frequency. To mitigate the restrictions with the waveguide simulator, and without the loss of generality, we compare the reflection characteristics of a square patch element for two different cases. 2.3.4.1  Comparison between PBC and Metallic Waveguides

In the first study, we compare the reflection coefficients of a square patch element as a function of frequency in a standard X‐band (WR‐90) waveguide. The inner dimensions of the waveguide are 22.86 × 10.6 mm; therefore, we select a unit‐cell size of 11.43 × 10.16 mm for the element that has an aspect ratio of 1.125. For the reflectarray element, a 62 mil Rogers 5880 laminate is selected for the substrate, and the square patch has a side length of 8 mm, which is the resonance dimension for the patch. Note that two elements are placed in the waveguide simulator as shown in Figure 2.12. The angle of incidence is frequency dependent and varies from 53° to 32° across 8.2–12.4 GHz, as shown in Figure  2.14. At 10 GHz the angle of incidence is 40.97°. Also note that the waveguide is excited in the fundamental mode, thus the incident wave is TE polarized. The simulation setup for the PBC method with Floquet port excitations carried out as described earlier. However, note that for this comparison one has to define the excitation angle in the simulations at each frequency accordingly. The simulations are carried out using Ansys HFSS, and reflection coefficients obtained using these two methods are given in Figure 2.15. It can be seen that a very good agreement is observed between these methods. It is important to note here that in reflectarray designs it is necessary to compute the reflection phase at a certain reference plane. Typically, in printed microstrip reflectarrays, the reference plane is the top surface of the patch, and one has to de‐embed the port in order to obtain the reflection phase at the desired reference plane.

55

Incident Wave (deg.)

50

45

40

35

30

9

10

11

12

Frequency (GHz)

Figure 2.14  The angle of incidence as a function of frequency in the standard X‐band waveguide.

23

Reflectarray Antennas

PBC with Floquet Port Waveguide Simulator

150 100

Γ (deg.)

50 0 –50 –100 –150 9

10

11

12

Frequency (GHz)

(a) 1

0.995

|Γ|

24

0.99

0.985 PBC with Floquet Port Waveguide Simulator 0.98

9

10 11 Frequency (GHz)

12

(b) Figure 2.15  Comparison between the reflection coefficients of an X‐band square patch reflectarray element using a waveguide simulator and a PBC model with Floquet port excitation: (a) phase, (b) magnitude.

2.3.4.2  Comparison between PBC and the Circuit Model

The comparative study presented here can also be conducted using the circuit model. However, since the excitation angle is a function of frequency, one has to derive the values of the lumped elements for each frequency independently. As such it will be more convenient to study a case where the angle of incidence is frequency independent. Here we study the reflection coefficients of the same element under normal incidence using the circuit model, and the PBC method with Floquet port excitation. As discussed

Analysis and Design of Reflectarray Elements

earlier, for the patch circuit model, one needs to determine the values of the capacitor, inductor and resistors. Typically, the conductor losses are less dominant in this frequency range and the majority of material losses is attributed to the dielectric substrate. In the circuit model for this element, the admittance of the parallel resistor is 14.286 μS. For the reactive elements, the values of capacitor and inductor are determined to be 0.1685pF and 1.4448nH, respectively. Comparison between the reflection coefficients obtained using these two methods are given in Figure 2.16, where again it can be seen that a very good agreement is observed between these methods. 150 PBC with Floquet Port Circuit Model

100

Γ (deg.)

50

0

–50

–100

–150

9

10

11

12

Frequency (GHz)

(a) 1.005 PBC with Floquet Port Circuit Model

|Γ|

1

0.995

0.99

0.985

9

10 11 Frequency (GHz)

12

(b) Figure 2.16  Comparison between the reflection coefficients of an X‐band square patch reflectarray element using a circuit model and a PBC model with Flouqet port excitation: (a) phase, (b) magnitude.

25

26

Reflectarray Antennas

In summary, a variety of techniques are available for analysis of reflectarray elements, which can accurately determine the reflection characteristics of the elements. While each of these methods have their own advantage, in general, the PBC method with Floquet port excitation is the most suitable approach for designing the element, since it can be conducted directly with a full‐wave simulator and also allows one to study the element properties under different excitation angles. The other two techniques are more suitable to use after the element has been designed. In particular, the advantage of the waveguide simulator is that the element characteristics can be obtained experimentally for verification purpose. The circuit model, on the other hand, provides a good insight into the physical characteristics of the element. In the next section, we will study a few classic examples of reflectarray elements.

2.4 ­Examples of Classic Reflectarray Elements In Section 2.2, three different methodologies were introduced which enable the reflectarray elements to achieve phase tuning. In this section, we provide one classic example for each of these phase tuning techniques, and outline the basic design principles. The unit‐cell analysis is carried out using the periodic boundary conditions with Floquet port excitation as described in Section 2.3. 2.4.1  Rectangular Patch with Phase‐Delay Lines In this approach, the patch element receives the space wave and converts the power into a guided wave that is transmitted through an attached stub. The signal reflects back from the stub termination, travels backward on the transmission line, and is then re‐radiated by the patch. Here we consider a Ka‐band microstrip reflectarray element, designed for 32 GHz operation. A 20 mil Rogers 5880 laminate is used for the element which has a dielectric constant of 2.2 and a loss tangent of 0.0009. The unit‐cell size is 4.7 × 4.7 mm2, which is about half‐wavelength at the center frequency. The design is carried out in two stages. First, a rectangular microstrip antenna element is designed which is excited by a microstrip feed line. The dimensions of the element are tuned to achieve an efficient radiation with a good impedance match at 32 GHz. According to reciprocal theory, it ensures that the signal received by the patch will be efficiently converted to a guided wave. The model of the element in Ansys HFSS along with the simulated reflection coefficient magnitude are given in Figure 2.17. The patch has length and width of 2.75 and 3.1 mm, respectively. The inset is 0.2 mm long, and 0.15 mm wide on each side of the feed line. The width of the feed line is 0.2 mm, which corresponds to an impedance of 140 Ω for the transmission line. It should be noted that unlike traditional array elements, that need to be designed for a certain input impedance, one typically has more freedom here, since in the reflectarray mode the line will be terminated with an open circuit. Also note that the reflection coefficient magnitude shown in Figure 2.17(b) is referenced to a 140 Ω port impedance. In the second stage, the element is studied for reflectarray operation, that is, the element is excited by a plane wave using a Floquet port. As discussed earlier, the phase shift provided by the delay line is proportional to twice the length of the line. As such, one needs to bend the line in order to realize a sufficient long physical length in the lattice.

Analysis and Design of Reflectarray Elements

(a) 0

|S11| (dB)

–5

–10

–15

–20 30

31

32

33

34

Frequency (GHz)

(b) Figure 2.17  A Ka‐band microstrip patch antenna array element: (a) 3D model of the element, (b) magnitude of reflection coefficient across the band.

The model of this reflectarray element in Ansys HFSS is shown in Figure 2.18, where the microstrip line with two mitered corner bends provides the necessary phase shift. The reflection coefficient of this Ka‐band reflectarray element is given in Figure 2.19, where it can be seen that this element yields an almost linear reflected phase variation as a function of the phase‐delay line length. With a total line length of 5.15 mm, the phase tuning range of this element is more than 450°. The average reflection coefficient magnitude is close to 0.98. The effects of the bends around 1.1 and 3.2 mm can be seen on the phase plot as small regions with almost constant phase. The design presented here can easily achieve a wider phase range by increasing the transmission line length; however, if several phase cycles are needed, it would be advantageous to use aperture coupled designs. More discussion on this will be given in

27

Figure 2.18  A Ka‐band microstrip reflectarray element with attached phase‐delay line stubs modeled in Ansys HFSS. 0

Γ (deg.)

–100

–200

–300

–400

–500

0

1

2

3

4

5

4

5

Length (mm)

(a) 1.01

1

|Γ|

0.99

0.98

0.97

0.96

0.95

0

1

2

3 Length (mm)

(b) Figure 2.19  Reflection coefficient of the Ka‐band microstrip reflectarray element as a function of the phase‐delay line length: (a) phase, (b) magnitude.

Analysis and Design of Reflectarray Elements

a later chapter. It is also interesting to compare the radiation characteristics of this element in antenna and reflectarray operation. The surface currents at 32 GHz are given in Figure 2.20, where it can be seen that while there is some difference in the current distribution on the microstrip lines, the surface current distribution on the patches have an almost identical form. As such, the element radiation patterns for these two modes are similar. The far‐field radiation pattern of the reflectarray element is given in Figure 2.21, where it can be seen that an almost omnidirectional radiation performance is obtained. More discussion on reflectarray element patterns and the effects of element beam‐width on the radiation patterns of the reflectarray will be given in a later chapter. 2

2 0.2 0.15

0 0.1 –1

0.2 y-axis (mm)

y-axis (mm)

1

0.05

–2 –2

0

0.15 0 0.1 –1

0.05

–2

0

2

1

–2

x-axis (mm)

0

2

x-axis (mm)

(a)

(b)

Figure 2.20  Magnitude of surface currents (|Jx| in A/mm) on the unit‐cells: (a) inset‐fed microstrip patch element, (b) reflectarray element with open‐circuit stub.

Z

dB(rETotal) 2.2500e + 001 2.0893e + 001 1.9286e + 001 1.7679e + 001 1.6071e + 001 1.4464e + 001 1.2857e + 001 1.1250e + 001 9.6429e + 000 8.0357e + 000 6.4286e + 000 4.8214e + 000 3.2143e + 000 1.6071e + 000 0.0000e + 000

X

Theta

Phi

Y

Figure 2.21  Far‐field pattern for a reflectarray element with an open‐circuit stub.

0

29

30

Reflectarray Antennas

2.4.2  Variable Size Square Patch The variable size square patch is perhaps the most popular element for reflectarrays. The design procedure is fairly simple, and the simplicity of the element shape is very advantageous for low‐cost fabrication, particularly at very high frequencies. In this design, the only parameter is the width of the patch. For a very small patch width, the reflection phase is basically that of the grounded dielectric slab which depends on the thickness of the laminate. The maximum patch width where the patch almost covers the entire unit‐cell essentially approaches that of an infinite conductor sheet, thus in the limit, the reflected phase is about −180°. The reflection phase varies with an S‐shaped curve as shown in Figure 2.8, from a maximum value to minimum as a function of the patch width. To be consistent with the previous design example, here we study a variable size square patch reflectarray element designed for 32 GHz operation, using the same material and lattice size. The patch width varies from 0.5 to 4.5 mm with a step size of 0.05 mm (about 2 mm), which is quite typical for low‐cost PCB fabrications. The element simulation model is given in Figure 2.22. The reflection coefficients of this Ka‐band reflectarray element are given in Figure 2.23, where it can be seen that this element can achieve a reflection phase range of more than 310°. While this element cannot realize a complete phase range of 360°, in most designs, a phase range above 300° for the elements is sufficient to yield a satisfactory performance with the reflectarray. Also it is important to point out that in general, elements with dimensions close to resonance (patch widths ranging from about 2–3.5 mm in this design) provide the largest variation in phase range, but they also exhibit the highest loss. Nonetheless, the minimum reflection coefficient magnitude is around 0.975 (about 0.2 dB loss), which is quite sufficient for the element. To get a better insight about the operating mechanism of the element, here we study the field properties of the resonant patch element which has a width of 2.73 mm. As discussed earlier, Floquet port excitation is used for the element analysis. The port is placed at a distance of 5 mm (about 0.53λ at 32 GHz) from the top surface of the element. For normal incidence, two polarizations can be excited; however, since the square patch is a dual‐linear element, these excitations are essentially identical. Here we consider the mode where the electric field is in the x‐direction (corresponding to the TM00 mode), and sample the fields at a distance of 0.1 mm (about 0.01λ at 32 GHz) from the top of the patch surface. The total electric fields are shown in Figure 2.24. Figure 2.22  A variable size Ka‐band microstrip reflectarray element modeled in Ansys HFSS.

Analysis and Design of Reflectarray Elements 150 100

Γ (deg.)

50 0 –50 –100 –150 –200

1

(a)

2 3 Patch Width (mm)

4

2

4

1.01

1

|Γ|

0.99

0.98

0.97

0.96

0.95

(b)

1

3

Patch Width (mm)

Figure 2.23  Reflection coefficients of the Ka‐band microstripreflectarray element as a function of the patch width: (a) phase, (b) magnitude.

The magnitude of the tangential Ex fields clearly show an electric field distribution that is identical to a microstrip square patch antenna that is excited in the fundamental mode, where the radiating slots are located at the two edges of the patch in the y‐direction. Similar to the patch antenna case, the cross‐polarized components, Ey, will cancel out in the principal planes since they are out of phase. The normal components, Ez, will also be shorted out on the conductive patch.

31

Reflectarray Antennas 40 30

1 0

20

–1

10

–2 –2

0

2

40

2 y-axis (mm)

y-axis (mm)

2

30

1 0

20

–1

10

–2

0

–2

x-axis (mm) 40

0

1

30

0

20

–1

10

–2 –2

0 x-axis (mm)

2

2 y-axis (mm)

y-axis (mm)

2

(b) 2

100

1

0

0 –1

–100

–2

0

(c)

–2

0 x-axis (mm)

2

(d) 2

2

0

0 –1

y-axis (mm)

100

1

–100

100

1 0

0

–1

–100

–2

–2 –2

(e)

0 x-axis (mm)

(a)

y-axis (mm)

32

0 x-axis (mm)

–2

2

0 x-axis (mm)

2

(f)

Figure 2.24  Total electric fields at a distance of 0.1 mm from the surface of the patch element: (a) |Ex|, (b) |Ey|, (c) |Ez|, (d) phase of Ex, (e) phase of Ey, and (f ) phase of Ez. The unit for the magnitude plots is V/mm, and the unit for phase plots is degrees.

In the case of reflectarrays, the primary aim of the unit‐cell analysis is to determine the reflection phase characteristic of the element. In this design, the co‐polarized fields are in the x‐direction. From Figure 2.24(d) it can be seen that the phase of the tangential Ex field is almost constant on its aperture. As discussed earlier, the element is excited with a plane wave, therefore, the magnitude and phase of the incident field is essentially

Analysis and Design of Reflectarray Elements 2 100

1

0

0 –1

y-axis (mm)

y-axis (mm)

2

–100

–2

0

0 –1

–100

–2 –2

0 x-axis (mm)

–2

2

(a)

0 x-axis (mm)

2

(b) 40

1

30

0

20

–1

10

–2 –2

0 x-axis (mm)

2

(c)

40

2 y-axis (mm)

2 y-axis (mm)

100

1

1

30

0

20

–1

10

–2

0

–2

0 x-axis (mm)

2

0

(d)

Figure 2.25  Phase of Ex: (a) incident fields, (b) scattered fields – amplitude of Ex, (c) incident fields, (d) scattered fields.

uniform on the element aperture. As such, the scattered field components would resemble the total fields. Nonetheless, the scattered fields can be computed directly using    Es Et Ei . (2.16) The phases of the incident and scattered Ex fields are given in Figure 2.25. As expected, the scattered fields produced by the reflectarray element, have an almost uniform phase on the aperture. This is basically the reflection phase obtained by the elements that are about 0° in this case for the resonant patch. It should be noted that in the port analysis, the reflection phase is averaged over the port surface, resulting in a single number for the reflection phase. The co‐polarized surface currents on the resonant patch, and the resulting far‐field electric fields are also shown in Figure 2.26, where it can be seen that the radiation pattern of the patch element is almost similar to the rectangular patch using the delay‐ line element studied earlier. 2.4.3  Single Slot Ring Elements The element rotation technique is specially devised for circular polarization applications. Square patches and the dual‐split ring geometry have been the most

33

Reflectarray Antennas 2 y-axis (mm)

34

0.4

1

0.3 0 0.2 –1 0.1 –2 –2

0

2

0

x-axis (mm)

(a) Z

dB(rETotal) 2.2500e + 001 2.0893e + 001 1.9286e + 001 1.7679e + 001 1.6071e + 001 1.4464e + 001 1.2857e + 001 1.1250e + 001 9.6429e + 000 8.0357e + 000 6.4286e + 000 4.8214e + 000 3.2143e + 000 1.6071e + 000 0.0000e + 000

(b)

X

Theta

Phi

Y

Figure 2.26  (a) Surface current magnitude (|Jx|) on the patch in A/mm, (b) Far‐field pattern.

frequently used elements. This phase tuning technique provides several advantages over the other two approaches. In particular, fabrication tolerances of elements are generally not as stringent as the other techniques. This is due to the frequency independent nature of this phase compensation scheme. On the other hand, one important consideration for the element rotation technique is the cross‐polarized component of the reflection coefficient, which in general is significantly affected by the incident angle. In this section, we study the performance of a Ka‐band reflectarray element using the element rotation technique. The element is a single slot ring (SSR). The geometrical model of the SSR element labeled with design parameters and the element simulation model in Ansys HFSS are shown in Figure 2.27.

Analysis and Design of Reflectarray Elements

θs

Rin

Rout

(a)

(b)

Figure 2.27  (a) Design parameters for an SSR element. (b) SSR element modeled in Ansys HFSS.

x

x x′ ψ

y

y

y′

Figure 2.28  Element rotation technique for SRR elements.

The element is designed for 32 GHz operation, and we use the same lattice size as the previous two examples. A 20 mil Rogers 5880 laminate is used for the element. The design parameters are Rin = 1.6 mm, Rout = 1.8 mm, and θs = 132.3°. Figure 2.29 (b). The element rotation technique is applied for phase control over the elements, as illustrated in Figure 2.28. The magnitudes of the element reflection coefficient for both co‐ and cross‐polarized components as a function of frequency are given in Figure 2.29, where we notice that at the center design frequency of 32 GHz, a perfect CP performance is obtained with this element. The element analysis is performed with a RHCP plane wave source. By rotating the element from −90° to +90°, one can obtain a full 360° phase tuning range. These results are given in Figure 2.30, where it can be seen that this element provides an almost ideal linear phase tuning as a function of rotation angle. As discussed earlier,

35

Reflectarray Antennas 1

Reflection Magnitude

0.8

0.6

0.4

0.2 co-polarization cross-polarization 0 30

31

32

34

33

Frequency (GHz)

Figure 2.29  Magnitude of the reflection coefficient as a function of frequency for the SSR element. 100

0

Γ (deg.)

36

–100

–200

–300

–50

0

50

Rotation Angle (deg.)

Figure 2.30  Reflection phase as a function of rotation angle for the SSR element.

theoretically there exists a linear dependence of the phase difference on the element rotation angle. Because the incident wave is RHCP, it is noted here that this curve is with a positive slope. The element rotation technique is a very practical approach for phase tuning; however, it is restricted to CP designs. At the center design frequency and for normal incidence, the cross‐polarized component of the reflection coefficient is almost zero. However, this value increases at off‐center frequencies, which limits the element bandwidth. More discussion on this topic will be given later on in this chapter.

Analysis and Design of Reflectarray Elements

2.5 ­Reflectarray Element Characteristics and Design Considerations In the previous sections, the basic design methodologies for reflectarray elements, along with the procedures for efficient design and analysis were outlined. Here we will look at some of the important characteristics of reflectarray elements. 2.5.1  Frequency Behavior of Element Reflection Coefficients The reflection coefficients of reflectarray elements are frequency dependent [23], [24]. As such, the element bandwidth is generally considered to be an important design parameter. The bandwidth of elements is usually related to the substrate thickness, and in most cases, a thicker substrate leads to a better bandwidth. However, depending on the phase tuning technique, different phenomena limit the bandwidth of the element. In the delay‐line technique, the element bandwidth is generally considered to be the return loss bandwidth, as shown in Figure 2.17. Outside this frequency band, the element will not receive the RF power efficiently, hence cannot provide the phase tuning capability. In addition to this, for phase‐delay lines, the phase shift provided by the line is also a function of frequency, which will result in phase errors at off‐center frequencies. For the element rotation technique, bandwidth is primarily limited by the tolerable level of the cross‐polarized component. While the desirable level or threshold depends on the design requirements, a practical definition of the element bandwidth for the element rotation technique is the range where the loss of the cross‐polarization level is at a tolerable level. This generally depends on the design. One option is to use the classic definition of 3 dB axial ratio bandwidth. Alternatively more stringent requirements for the array may limit the cross‐polarization to be 20 dB below the co‐polarization level. For the variable element size technique, bandwidth definition is a bit more complicated. While some figures of merit have been proposed [25, 26], in general the linearity of the phase curve, and curves with small slopes, prove to be a practical measure to analyze the bandwidth performance of the element. Since the phase curve relies on the element size, whereas the size is linked to the resonant frequency, the phase curve implies the element phase stability with respect to frequency variation. Unlike the phase‐delay line technique, there is no matching problem with the variable element size technique. As discussed earlier, the variable size technique is the most popular choice for reflectarray elements. Therefore in this section we will take a closer look at the frequency response of this technique. The variable size square patch element designed in the previous section is considered for this study. The reflection coefficients at three different frequencies as a function of patch width are given in Figure 2.31. It can be seen that as expected, as the frequency increases, the size of the resonance patch decreases. This non‐linear variation in phase as a function of frequency is the primary reason for the narrow bandwidth of reflectarrays with variable size patch elements. Several broadband techniques, however, exist that can mitigate this problem. These will be discussed in Chapters 6 and 7. 2.5.2  Effects of Oblique Incidence Angles on Element Reflection Coefficients In many cases reflectarray elements are designed under normal incidence approximation. In practice however, elements on the aperture of the reflectarray will be excited

37

Reflectarray Antennas 150 30 GHz 100

32 GHz 34 GHz

Γ (deg.)

50 0 –50 –100 –150 –200

1

2

3

4

Patch Width (mm)

(a) 1.01

1

0.99

|Γ|

38

0.98

0.97 30 GHz 0.96

32 GHz 34 GHz

0.95

1

2

3

4

Patch Width (mm)

(b) Figure 2.31  Reflection coefficients of the variable size square patch reflectarray element as a function of the patch width at three frequencies across the band: (a) phase, (b) magnitude.

with different oblique angles. While in many cases, especially for thin substrates, the effect of the oblique excitation angle on the reflection coefficients is small, depending on the design, one may need to consider this issue. The effects of oblique excitation angles generally depends on the choice of phase tuning method. For elements with delay lines, typically oblique excitation only manifests as a reduced received power for the element, the level of which depends on the element pattern. For the rotated element

Analysis and Design of Reflectarray Elements

technique, the effect is primarily an increase in the level of cross‐polarized component. Effects of oblique angles of incidence are more complicated for variable size elements, since both phase and magnitude of the element are dependent on the excitation angle. As an example, here we will study the effects of oblique incident angles for the variable size square patch element designed in the previous section. The reflection coefficients are shown in Figure 2.32 and Figure 2.33 for several different excitation angles, for perpendicular (or H) and parallel (or V) polarizations, respectively. 200

Γ (deg.)

100

0

–100

–200

θi = 0° θi = 10° θi = 20° θi = 30° θi = 40° 1

2

3

4

Patch Width (mm)

(a) 1

|Γ|

0.95

0.9

0.85

θi = 0° θi = 10° θi = 20° θi = 30° θi = 40° 1

2

3

4

Patch Width (mm)

(b) Figure 2.32  Reflection coefficients of the variable size square patch reflectarray element as a function of the patch width for different angles of incidence with perpendicular polarization: (a) reflection phase for φ = 0°, (b) (a) reflection magnitude for φ = 0°, (c) reflection phase for φ = 45°, (c) and (d) reflection magnitude for φ = 45°.

39

Reflectarray Antennas 200 θi = 0° θi = 10° θi = 20° θi = 30° θi = 40°

Γ (deg.)

100

0

–100

–200

1

2 3 Patch Width (mm)

4

2

4

(c) 1

0.99 |Γ|

40

0.98

0.97

θi = 0° θi = 10° θi = 20° θi = 30° θi = 40° 1

(d)

3

Patch Width (mm)

Figure 2.32  (Continued)

It can be seen that, as expected, while the effects of oblique excitation are not significant for most cases, they result in both phase and magnitude errors in the reflection coefficient. This may be accounted for in the design by generating data tables to select the appropriate element size based on the position of the element on the aperture and its true angle of incidence [2]. It is also important to note that for large patch sizes, that is, when the patch covers most of the lattice, higher order resonances are observed for oblique angles. As such depending on the design, one may need to avoid using these larger patches.

Analysis and Design of Reflectarray Elements

2.5.3  Sources of Phase Error in Reflectarray Element Design The phasing elements are a key component in the reflectarray antenna design. Therefore, it is important to note that the accuracy of the unit‐cell data used for the design is susceptible to errors arising from the design method, fabrication, or approximations in the analysis. A brief description of each of these element errors is given here. 200

Γ (deg.)

100

0

–100

–200

θi = 0° θi = 10° θi = 20° θi = 30° θi = 40° 1

2

3

4

Patch Width (mm)

(a) 1

|Γ|

0.99

0.98

0.97

θi = 0° θi = 10° θi = 20° θi = 30° θi = 40° 1

2

3

4

Patch Width (mm) (b)

Figure 2.33  Reflection coefficients of the variable size square patch reflectarray element as a function of the patch width for different angles of incidence with parallel polarization: (a) reflection phase for φ = 0°, (b) (a) reflection magnitude for φ = 0°, (c) reflection phase for φ = 45°, (c) and (d) reflection magnitude for φ = 45°.

41

Reflectarray Antennas 200

Γ (deg.)

100

0

–100

–200

θi = 0° θi = 10° θi = 20° θi = 30° θi = 40° 1

2

3

4

Patch Width (mm) (c) 1

0.95

|Γ|

42

0.9

0.85

0.8

θi = 0° θi = 10° θi = 20° θi = 30° θi = 40° 1

2

3

4

Patch Width (mm) (d)

Figure 2.33  (Continued)

Quantization Errors: In practice, fabrication accuracy plays a significant role in the element’s performance. The dimensions or rotations of the elements are changed by a certain amount depending on the fabrication precision, and as such, a continuous phase control is not possible. For each element on the aperture, a phasing element is selected that provides the closest quantized phase with respect to the ideal phase shift. The difference between the ideal element phase and the quantized phase of the selected element is categorized as a quantization error. Phase Range Errors: Another important factor in element selection is the available phase range of the elements. Typically single‐resonant phasing elements have a phase

Analysis and Design of Reflectarray Elements

range below 360°. Although this problem can be avoided by using multi‐resonance elements, if the phase range is smaller than the complete cycle, some elements will inevitably have unattainable phase shifts. While the element selection routine minimizes these errors by selecting the closest quantized values, these errors are in nature different from the quantization errors and are categorized as errors due to limited phase range. Infinite Array Approximation: As discussed earlier, the infinite array approach is used to obtain the unit‐cell data, which assumes that the dimensions (or rotations) of the adjacent elements should be identical or at least very close. In conventional single‐beam reflectarrays with variable size elements, the required phase, and therefore dimensions of adjacent elements, grow gradually in each Fresnel zone. When the zone is complete, a phase wrap is observed and the element dimension jumps from a maximum size to a minimum size or vice versa, which violates the periodic approximation and results in some error. Similar conditions are observed for other phase control schemes.

2.6 ­Reflectarray Element Measurements As discussed earlier in this chapter, one of the great advantages of the metallic waveguide simulation technique is the fact that it can be verified experimentally. In this section, we will present experimental and simulated results for a variable size square patch reflectarray element using a WR75 standard waveguide. An image of the components used for the experiment is given in Figure 2.34, where the reflectarray element is placed in a

Waveguide (WR 75)

coaxial-toCover board waveguide adapter (two layer)

Reflectarray element (square patch)

Figure 2.34  The waveguide components for reflectarray element measurements.

43

44

Reflectarray Antennas incident wave illuminate upper patch

incident wave illuminate ground plane

Figure 2.35  Measurement setups for the reflectarray element.

cover board for measurements. In this setup, when the element is placed such that the incident wave illuminates the patch, the element loss can be measured. Alternatively, when the element is placed such that the incident wave illuminates the ground plane, the ground plane loss can be measured. Measurement setups for the reflectarray element when the incident wave illuminates the upper patch and ground plane are shown in Figure 2.35. A two‐port waveguide calibration technique is used to calibrate the systems at the waveguide plane. The measurement procedure used in this experiment is summarized here. 1) Use two‐port waveguide calibration to calibrate at the waveguide plane. 2) Insert a reflectarray element sample into the cover board, connect it to the waveguide, and measure reflection coefficient of the element (phase and loss). The reflection phase of the element is obtained in this stage. 3) Insert the reflectarray element into cover board reversely, connect it to the waveguide, and measure the ground plane loss. The loss of the patch element is the difference between losses measured in step 2 and step 3. The element sample in the study presented here is fabricated on a 62 mil Taconic TLX‐8 laminate. The substrate permittivity and loss tangent are 2.55 and 0.0019, respectively. The element periodicity is 9.52 mm. Measured and simulated reflection coefficients for two different patch sizes are given in Figure 2.36. Note that a good agreement between measured and simulated reflection phase is observed in these results and the maximum phase difference is less than 10°. On the other hand measured

Analysis and Design of Reflectarray Elements

and simulated losses show a larger discrepancy. This is primarily due to the fact that several factors such as stability of the coaxial line, the gap between cover board and waveguide, the gap between the element and cover board, and so on, influence the losses in the system. In general element loss is relatively hard to measure, particularly at higher frequencies. Nonetheless, the difference between measured and simulated losses is less than 0.3 dB at 12.5 GHz, and no more than 0.4 dB across the 11.5–13.5 GHz frequency range.

Patch size = 5.3 mm 200 150

Reflected phase

100 50 0 –50 –100 –150 –200 1.15

measured result simulated result 1.2

1.25

1.3

1.35

Frequency (GHz) Patch size = 6 mm

200 150

Reflected phase

100 50 0 –50 –100 –150 –200 11.5

(a)

measured result simulated result 12

12.5

13

13.5

Frequency (GHz)

Figure 2.36  Measured and simulated reflection coefficients for variable size reflectarray elements: (a) element phase, (b) element loss.

45

Reflectarray Antennas Patch size = 5.3 mm

0.3 0.2

Reflected loss

0.1 0 –0.1 –0.2 –0.3 –0.4 –0.5 11.5

measured result simulated result 12

12.5

13

13.5

Frequency (GHz) Patch size = 6 mm

0.3 0.2 0.1 Reflected loss

46

0 –0.1 –0.2 –0.3 –0.4 –0.5 11.5

measured result simulated result 12

12.5

13

13.5

Frequency (GHz)

(b) Figure 2.36  (Continued)

­References 1 D. G. Berry, R. G. Malech, and W. A. Kennedy, “The reflectarray antenna,” IEEE

Transactions on Antennas and Propagation, Vol. 11, No. 6, pp. 645–651, Nov., 1963.

2 J. Huang and J. A. Encinar, Reflectarray Antennas, IEEE press by John Wiley & Sons

Ltd. 2008.

3 R. Munson, H. A. Haddad, and J. W. Hanlen, “Microstrip reflectarray for satellite

communication and radar cross‐section enhancement or reduction,” U.S. patent 4684952, Washington D. C., Aug. 1987. 4 R. J. Mailloux, Phased Array Antenna Handbook, 2nd Edn, Artech House, 2005.

Analysis and Design of Reflectarray Elements

5 M. I. Skolnik, Radar Handbook, 3rd Edn, McGraw‐Hill, 2008. M. E. Cooley, and D. Davis, “Reflector antennas,” in Radar Handbook, 3rd Edn, McGraw‐ 6

Hill, 2008.

7 D. C. Chang and M. C. Huang, “Multiple‐polarization microstrip reflectarray antenna

8

9

10

11 12

13 14

15

16 17 18 19 20 21

22 23

24

with high efficiency and low cross‐polarization,” IEEE Transactions on Antennas and Propagation, Vol. 43, No. 8, pp. 829–834, Aug. 1995. R. D. Javor, X. Wu, and K. Chang “Design and performance of a microstrip reflectarray antenna,” IEEE Trans. Antennas and Propagation, Vol. 43, No. 9, pp. 932–939, Sept., 1995. K. Chen, C. C. Tzuang, and J. Huang, “A higher order microstrip reflectarray at Ka‐band,” in IEEE Antennas and Propagation Society International Symposium Digest, pp. 566–569, July, 2001. D. G. Gonzalez, G. E. Pollon, and J. F. Walker, “Microwave phasing structures for electromagnetically emulating refl ective surfaces and focusing elements of selected geometry,” Patent US 4905014, Feb. 1990. D. Pozar and T. A. Metzler, “Analysis of a reflectarray antenna using microstrip patches of variable size,” IEE Electronics Letters, Vol. 29, No. 8, pp. 657–658, Apr. 1993. M. E. Bialkowski and K. H. Sayidmarie, “Investigations into phase characteristics of a single‐layer reflectarray employing patch or ring elements of variable size,” IEEE Trans. Antennas and Propagation, Vol. 56, No. 11, pp. 3366–3372, Nov, 2008. H. R. Phelan, “Spiralphase reflectarray for multitarget radar,” Microwave Journal, Vol. 20, pp. 67–73. July 1977. J. Huang and R. J. Pogorzelski, “A Ka‐band microstrip reflectarray with variable rotation angles,” IEEE Trans. Antennas and Propagation, Vol. 46, No. 5, pp. 650–656, May, 1998. H. Rajagopalan and Y. Rahmat‐Samii, “On the reflection characteristics of a reflectarray element with low‐loss and high‐loss substrates,” IEEE Trans. Antennas and Propagation, Vol. 52, No. 4, pp. 73–89, Apr. 2010. P. W. Hannan and M. A. Balfour, “Simulation of a phased‐array antenna in waveguide,” IEEE Trans. Antennas and Propagation, Vol. 13, No. 3, pp. 342–353, May, 1965. A. K. Bhattacharyya, Phased Array Antennas: Floquet Analysis, Synthesis, BFNs and Active Array Systems, John Wiley & Sons, Ltd, 2006. ANSYS HFSS, ANSYS Inc. Website available at /www.ansys.com/ (accessed July 2017). CST Microwave Studio, CST Studio Suite, Computer Simulation Technology AG. Website available at https://www.cst.com/ (accessed July 2017). FEKO, Altair Engineering Inc. Website available at https://www.feko.info/ (accessed July 2017). H. Mosallaei and K. Sarabandi, “Antenna miniaturization and bandwidth enhancement using a reactive impedance substrate,” IEEE Trans. Antennas Propag., Vol. 52, No. 9, pp. 2403–2414, Sep. 2004. M. Bozzi, S. Germani, and L. Peregrini, “Modeling of losses in printed reflectarray elements,” European Microwave Conference, Oct 2004. J. Huang, “Bandwidth study of microstrip reflectarray and a novel phased reflectarray concept,” in IEEE Antennas and Propagation Society International Symposium Digest, Vol. 1, pp. 582–585, July 1995. D. Pozar, “Bandwidth of reflectarrays,” IEE Electronics Letters, Vol. 39, No. 21, pp. 1490–1491, Oct. 2003.

47

48

Reflectarray Antennas

25 M. Bozzi, S. Germani, and L. Perregrini, “Performance comparison of different element

shapes used in printed reflectarrays,” IEEE Antennas and Wireless Propagation Letters, Vol. 2, pp. 219–222, 2003. 6 H. Rajagopalan and Y. Rahmat‐Samii, “Dielectric and conductor loss qualification for 2 microstrip reflectarray: simulations and measurements,” IEEE Trans. Antennas and Propagation, Vol. 56, No. 4, pp. 1192–1196, Apr. 2008.

49

3 System Design and Aperture Efficiency Analysis Design of a reflectarray antenna typically consists of two steps, namely, the phasing element design and the system configuration design. In the previous chapter, the analysis of phasing elements is presented, and this chapter focuses on the analysis of the reflectarray system configuration. Usually, the system design begins with a specified gain requirement. Similar to traditional reflector antennas [1], [2], the gain of a reflectarray antenna is proportional to the electrical size of the aperture, and is calculated as the product of the aperture efficiency and maximum aperture directivity [3], [4]. Through the aperture efficiency analysis, one could estimate the required size of the reflectarray antenna aperture in order to achieve the desired gain. More importantly, it is desirable to design the system such that the aperture efficiency is maximized. Through the aperture efficiency analysis, one could optimize the parameters of a system configuration, such as feed specifications, f/D ratio, aperture shape, and so on. Hence, aperture efficiency analysis is critical in the system design. In addition to aperture efficiency considerations, several other factors influence the system design, including feed blockage and edge taper. Feed blockage is generally minimized by offsetting the feed antenna, which usually decreases the aperture efficiency and increases the cross‐polarization. Typically, edge taper has to be kept below a certain level to ensure that strong diffractions don’t occur at the reflectarray edge. Consequently, in order to realize an overall optimum performance some compromise has to be made in the system design. In this chapter, we will study the basic considerations for an efficient reflectarray system design. Generalized formulations for computing the aperture efficiency and taper of planar reflectarray antennas are outlined, and several reflectarray system designs are analyzed. Based on these studies some general guidelines for reflectarray design are presented.

3.1 ­A General Feed Model In reflectarray antenna systems, the feed antenna position and radiation characteristics are one of the most influential measures in the design. The aperture illumination is directly dependent on the radiation pattern of the feed antenna and its position, as such, the efficiency and other system characteristics of the reflectarray antenna cannot be Reflectarray Antennas: Theory, Designs, and Applications, First Edition. Payam Nayeri, Fan Yang, and Atef Z. Elsherbeni. © 2018 John Wiley & Sons Ltd. Published 2018 by John Wiley & Sons Ltd.

50

Reflectarray Antennas Feed Antenna Phase Center Feed P.P.1 Feed P.P.2

Figure 3.1  Coordinate system of the reflectarray and feed system.

Z

XF YF ZF X

Y rf

determined without a proper description of the feed characteristics. In this section, we will present a generalized model for the radiation pattern of the feed antenna. Based on this feed model, we will carry out aperture efficiency analysis in the following sections. Without the loss of generality, let’s consider the geometrical setup of the reflectarray system shown in Figure 3.1. The reflectarray aperture is lying on the xy‐plane and the aperture center is located at the center of the coordinate system. The feed antenna is pointing toward the geometrical center of the aperture. With the geometrical system setup specified, if one can now describe the radiation pattern of the feed antenna with a closed form mathematical expression, it would then be possible to analyze the system efficiency directly. 3.1.1  Models of Linearly Polarized and Circularly Polarized Feeds Various pattern models have been developed to simulate the radiation properties of reflector feed antennas. Among these models, the cosq pattern is perhaps the most common approach, because of its simplicity and versatility to describe different pattern types [3], [5]. In this model, the radiation pattern of the ideal feed antenna with a fixed phase center is typically described by the two principal plane patterns, and is given by − jkr f



e E F ( r f ,θ f ,ϕ f ) = A0 θˆC E (θ f ) cosϕ f − ϕˆC H (θ f ) sin ϕ f    rf



e E F ( r f ,θ f ,ϕ f ) = A0 θˆC E (θ f ) sin ϕ f + ϕˆC H (θ f ) cosϕ f    rf

− jkr f

, for x polarized, (3.1)

, for y polarized. (3.2)

Here CE (θf) and CH (θf) are the radiation patterns of the feed antenna in the E‐ and H‐planes, respectively, and rf is the vector from the feed phase center (focal point) to any point on the reflectarray aperture. Note that E‐ and H‐planes depend on the orientation and polarization of the feed antenna, and correspond to the feed principal planes, P.P.1 or P.P.2 in Figure 3.1, accordingly. Also note that θf and φf are the spherical angles of the feed coordinate system, that is (XF, YF, ZF) shown in Figure 3.1. In this model, the

System Design and Aperture Efficiency Analysis

shape of the feed radiation pattern is given in terms of the feed elevation angle (θf) and a power factor q, which in general is different for E‐ and H‐planes, that is,

CE (

f

) cos q E (



CH (

f

) cos q H (

f f

) E‐plane pattern of the feed antenna,

(3.3)

) H‐plane pattern of the feed antenna

(3.4)

Note that the back radiation of the feed pattern is assumed to be zero in these equations. The shape of the pattern is controlled by qE and qH, which can be determined by matching CE and CH to the measured pattern of the feed antenna. It should be noted that with this feed model, for any azimuth angle other than φf = 0° and 90°, the field from the feed is approximated by interpolation. As a simple example, first we consider a short dipole antenna, which is centered at the phase center point shown in Figure 3.1 and oriented along XF. The feed principal planes (Feed P.P.1 and Feed P.P.2 in Figure 3.1) depend on the orientation of the feed antenna. With this setup, the E‐plane is P.P.1 and the H‐plane is P.P.2 which are given by

CE

f

cos

f

,C H

1, for an x f -polarized short dipole. (3.5)

f

Besides the linearly polarized feeds, circularly polarized (CP) feed patterns can also be represented with this model [5], which is given by e − jkrf E F ( r f ,θ f ,ϕ f ) = A0 e jτϕ f θˆC E (θ f ) + ϕˆ jτ C H (θ f )  , for circularly polarized. (3.6)   rf For the circular polarized feed patterns, the parameter τ determines the sense of polarization. For a left‐handed (LH) CP feed τ = +1 and for right‐handed (RH) CP feed τ = −1. Note that this equation represents a perfect circular polarized wave only in the main beam direction (θf = 0) of the feed coordinate system, unless C E ( f ) C H ( f ). It is important to note here that for circular polarization, E‐ and H‐planes cannot be defined, and CE and CH simply refer to the patterns in xz and yz planes, respectively. 3.1.2  Balanced Feed Models In the dipole feed example, qE and qH are 1 and 0, respectively, and the pattern is asymmetric. For reflector feeds, the asymmetric pattern is not desirable in most cases, and rotationally symmetric patterns are often desired, that is,

CE

f

CH

f

C

f

cos q

f

. (3.7)

Feed antennas that exhibit rotationally symmetric patterns are referred to as balanced feeds. With the cosq model, this means that qE = qH = q,



E F ( r f ,θ f ,ϕ f ) = A0

e − jkrf C (θ f ) θˆ cosϕ f − ϕˆ sin ϕ f  , for x polarized, (3.8) rf

E F ( r f ,θ f ,ϕ f ) = A0

e − jkrf C (θ f ) θˆ sin ϕ f + ϕˆ cosϕ f  , for y polarized, (3.9)   rf

E F ( r f ,θ f ,ϕ f ) = A0

e − jkrf C (θ f ) e jτϕ f θˆ + ϕˆ jτ  , for circularly polarized. (3.10) rf

51

Reflectarray Antennas

For the balanced feed described in this section, the normalized power pattern of the feed antenna is given by PF

f ,

cos2 q 0

f



f

for 0 elsewhere

/2

.

(3.11)

With this feed radiation model, the feed antenna directivity and pattern shape are described by a single parameter q. Note that the back radiation of the feed pattern is assumed to be zero in this equation. The radiation pattern for this feed model is shown in Figure  3.2 for a few different values of q, where it can be seen that as q increases, the beamwidth of the feed pattern decreases. Consequently, the antenna directivity increases as the value of q is increased. The directivity as a function of the q value is given in Figure 3.3.

Radiation Pattern (dB)

Figure 3.2  Radiation pattern of the cosq feed pattern model for different values of q.

q=2

0

q=6 q = 18

–5 –10 –15 –20

–50

0

50

θf (degrees)

Figure 3.3  Directivity of the cosq feed pattern model as a function of q.

20

15 Directivity (dB)

52

10

5

0

0

5

10 q

15

20

System Design and Aperture Efficiency Analysis

3.2 ­Aperture Efficiency In aperture type antennas such as reflectarrays, the physical aperture area A determines the theoretical maximum directivity of the antenna, that is,

Dmax

A

4

2

. (3.12)

Here λ is the wavelength at the design frequency. The reflectarray antenna gain can then be computed as

G

A

4

2

aperture , (3.13)

where ηaperture is the aperture efficiency of the antenna. Similar to reflector antennas, many efficiency factors influence the overall aperture efficiency of reflectarray antennas, and the dominant terms are typically spillover (ηs) and illumination (ηi) efficiencies. In this section, we study the aperture efficiency as the product of these two terms,

s i . (3.14)

aperture

It is important to note here that these two important efficiency factors are directly related to the antenna configuration parameters, hence they also serve as key factors for the reflectarray system design. Other efficiency factors are typically associated with the choice of reflectarray phasing elements such as material losses, losses due to phase error, mismatch, and polarization losses. Further discussion on these losses and their influence on aperture efficiency will be given when we discuss loss budget calculations in Chapter 4. 3.2.1  Spillover Efficiency Spillover efficiency (ηs) is defined as the percentage of radiated power from the feed that is intercepted by the reflecting aperture relative to the total radiated power. Mathematically, this is given by  

ηs =



∫∫ P(r f ) ⋅ ds A

∫∫

Sphere

   P (r f ) ⋅ ds (3.15)

where both integrals are fluxes of the Poynting vector, P(r), evaluated over certain surfaces. The denominator is the total power radiated by the feed; therefore, the integral is performed over the entire surface area of a virtual spherical surface that encapsulates the feed antenna and is centered at the phase center of the feed. The integral of the numerator, the intercepted power, is evaluated over the aperture of the reflectarray antenna, as shown in Figure 3.4.

53

54

Reflectarray Antennas

Figure 3.4  Geometrical setup of the feed and the reflectarray aperture.

Sphere Z (xf, yf, zf)

A θ0

H

rf Y

θp X

Using the cosq pattern model for a balanced feed, the Poynting vector of the feed antenna is defined as



  cos2 q (θ f P ( r f ) = rˆf r f2

)

for 0 ≤ θ f ≤ π /2. (3.16)

The denominator in (3.15) can now be computed quite easily, and will have a simple closed form expression given by



   P r f .ds

2

/2

cos2 q

0 0

Sphere

f

sin

f

d

f

d

f

2 . (3.17) 2q 1

The numerator in (3.15) is dependent on the feed position and aperture shape, and is typically computed numerically [6]. The geometrical setup of the system is shown in Figure 3.4. 3.2.2  Illumination Efficiency Similar to spillover efficiency, the definition of illumination efficiency for reflectarrays is also extended from reflector antennas, which is mathematically given as 2

i



1 Aa

I A dA A

I A

2

dA

. (3.18)

A

Here I is the amplitude distribution over the reflectarray aperture, A. As discussed earlier, in the analysis outlined here it is assumed that the feed is purely in a certain polarization. The amplitude on the aperture is then a function of the feed (transmit) and reflectarray element (receive) patterns. For reflectarrays, the radiation pattern of the element can also be modeled with a cosq pattern, and to simplify the model, we assume that the element also has a rotationally

System Design and Aperture Efficiency Analysis

symmetric pattern. To distinguish between the feed and element pattern power factor, we use qe for the element pattern. The mathematical model of this element pattern is given by UE

p,

cos2 qe 0

p



for 0 p elsewhere

p

/2

. (3.19)

Here θp is the elevation angle in the local coordinate system for any point on the reflectarray aperture as shown in Figure 3.4. Typically, the values of qe for reflectarray elements are much smaller than the values of q, in other words, the element pattern has a broader beamwidth. The nominal value for qe is about 1. Note that this value of qe corresponds to a directivity of 7.78 dB, which is typical for microstrip patches that are conventionally used for reflectarray phasing elements. Note that in some engineering designs, qe is set to zero for simplicity. The normalized amplitude at any point of the reflectarray aperture can now be expressed in a simple form, which is given by



In A

cos q rf

f

cos qe

p

. (3.20)

Here the first term represents the amplitude on a certain point on the aperture coming from the feed antenna, thus the denominator rf is added to account for the path length, since rays leaving the feed at the focal point spread out in all directions with a 1/rf variation. It should be noted here that since the element receiving pattern is included in this calculation, the energy radiated from the aperture is smaller than the energy illuminated on the aperture. With the amplitude distribution defined on the aperture, the illumination efficiency can be computed numerically. Note that when computing (3.20), the angles θf and θp are computed for every point on the aperture in terms of the feed coordinate system, Figure 3.1, and the local point coordinate system, Figure 3.4, respectively. 3.2.3  Effects of Aperture Shape on Efficiency Similar to reflector antennas, the optimum aperture efficiency for a reflectarray is achieved when the feed pattern is matched to the reflectarray [3]. Consequently, the shape of the reflectarray aperture, the radiation pattern of the feed antenna, and the feed location play pivotal roles in yielding maximum aperture efficiency for the reflectarray system. Here we will compare the aperture efficiencies of a few typical reflectarray aperture configurations when a balanced feed with a rotationally symmetric pattern is used as the primary feed antenna [3]. A front‐fed reflectarray configuration is considered here, where the feed antenna is placed at a distance of 18λ from the center of the aperture. Three different aperture shapes are studied here, where in each case, the power pattern of the feed (the value of q) is selected such that the reflectarray achieves its maximum aperture efficiency. For the first case, we consider a 20λ × 30λ rectangular aperture for the reflectarray. Aperture efficiency as a function of q is given in Figure 3.5(a), where the optimum

55

Reflectarray Antennas

100 ηi ηs ηa

Efficiency (%)

90 80 70 60 50 40

5

10

(a)

15

q

1

10 5 y (λ)

56

0.5

0 –5 –10

(b)

–10

0

10

0

x (λ)

Figure 3.5  Aperture efficiency for a reflectarray with a rectangular aperture: (a) efficiency versus q, (b) normalized amplitude on the aperture for q = 4.5.

value is 4.5. Here the spillover and illumination efficiencies are computed using (3.15) and (3.18), respectively. The total aperture efficiency is the product of these two terms, which corresponds to an aperture efficiency of 69.76%. The normalized amplitude distribution on the aperture for this q value is also given in Figure 3.5(b). It can be seen that for this aperture shape, even with an optimum feed pattern, the wider side of the reflectarray aperture is not properly illuminated, which leads to a low aperture efficiency. In general for these types of apertures, a feed antenna that can provide a pattern with different beamwidths in the two orthogonal directions proportional to the x/y ratio of the aperture, can significantly improve the aperture efficiency of the reflectarray system. However, such a reflectarray design will have a non‐symmetric main beam that is broader in the direction of the smaller side of the aperture. For the same system setup, now let us consider a square aperture with a side length of 20λ for the reflectarray. Similarly, aperture efficiency as a function of q is given in Figure 3.6(a). The optimum value of q is 6.5 in this case, which corresponds to an aperture efficiency of 75.35%. The normalized amplitude distribution on the aperture for

System Design and Aperture Efficiency Analysis 100 90 Efficiency (%)

80 70 60 ηi

50

ηs

40 30

(a)

ηa 5

10

15

q

1

10

y (λ)

5

0.5

0

–5

–10 –10

(b)

0

10

0

x (λ)

Figure 3.6  Aperture efficiency for a reflectarray with a square aperture: (a) efficiency versus q, (b) normalized amplitude on the aperture for q = 6.5.

this value of q is also given in Figure 3.6(b), where it can be seen that in comparison with the rectangular aperture studied earlier, a much better illumination is achieved on the aperture, which ultimately results in a better aperture efficiency. However, note that the corners of the square are still not properly illuminated. This is again due to the fact that the spherical wave front of the symmetric feed pattern cannot be matched to a square aperture shape. The previous two cases clearly show the influence of aperture shape on the efficiency of reflectarray systems. For maximum efficiency, the shape of the aperture should directly correlate with the shape of the feed radiation pattern. Consequently, for a balanced feed pattern, the ideal aperture shape is circular. To verify this, here we study a reflectarray antenna with a circular aperture with a diameter of 20λ for the same system setup. For this aperture, the optimum value of q is 8.5, which corresponds to

57

Reflectarray Antennas 100 90 Efficiency (%)

80 70 60 ηi

50

ηs

40 30

(a)

ηa 5

10

15

q

1

10

5

y (λ)

58

0

0.5

–5

–10 –10

(b)

0 x (λ)

10

0

Figure 3.7  Aperture efficiency for a reflectarray with a circular aperture: (a) efficiency versus q, (b) normalized amplitude on the aperture for q = 8.5.

an aperture efficiency of 78.12%, and is notably higher than the previous two cases. Aperture efficiency as a function of q and the normalized amplitude distribution on the aperture for the optimum q value are both given in Figure 3.7. The rotationally symmetric amplitude taper on the reflectarray aperture, Figure 3.7(b), also leads to a desirable radiation pattern for the reflectarray, and in general, a circular aperture is the optimum aperture shape for front‐fed symmetric reflectarrays. For offset configurations, however, the asymmetry of the system results in a non‐symmetric amplitude taper that degrades the efficiency. Elliptic aperture shapes may yield a better performance for offset configurations, however, for small feed offset angles, which is usually the case for reflectarrays, the circular aperture shape will yield good aperture efficiency.

System Design and Aperture Efficiency Analysis

3.2.4  Effects of Feed Location on Efficiency In the previous aperture shape study, the feed location is fixed. In practical reflectarray designs, the feed location is an important system design parameter to design. For this setup, the two parameters in the design are the feed radiation pattern model, q, and the feed distance from the reflectarray aperture (z). Aperture efficiency as a function of both these parameters is given in Figure  3.8(a). It can be seen that for different feed patterns, the aperture efficiency can be maximized by selecting different feed positions. In general, the larger the q factor, the higher the feed position and the higher the aperture efficiency. Very broad feed patterns (small values of q) typically cannot yield high aperture efficiency. On the other hand, while very directive feed patterns (large values

14

70

12 60

q

10 50

8 6

40

4

30

2 10

15

20

25

20

z (λ)

(a)

100

Efficiency (%)

90 80 ηi

70

ηs ηa

60 50 10 (b)

15

20

25

z (λ)

Figure 3.8  Efficiency for a reflectarray with a front‐fed symmetric reflectarray: (a) aperture efficiency as a function of feed position and q, (b) efficiency as a function of feed position for q = 6.5.

59

60

Reflectarray Antennas

of q) can attain high aperture efficiency, they usually are not considered a good choice for the reflectarray since the corresponding feed antennas typically have a large physical volume (resulting in high aperture blockage), and also reduce the relative gain improvement achieved by the reflector system. Values of q in the range of 6–8 are typically good choices. For the same system, efficiency as a function of feed position is given for a feed with q = 6.5 in Figure 3.8(b). Note that illumination efficiency increases as the feed distance to aperture is increased, while the spillover efficiency decreases. Consequently, the optimum position is a nominal value in between, which is 16.3λ for this design and corresponds to an aperture efficiency of 77.34%.

3.3 ­Aperture Blockage and Edge Diffraction 3.3.1  Aperture Blockage and Offset Systems Aperture blockage, mainly the feed blockage, is a physical characteristic that generally exists in reflectors and reflectarrays, where the feed antenna (and the support structure) blocks some of the radiation, as schematically shown in Figure 3.9(a). In general, aperture blockage is a function of the ratio of the blockage diameter (D0) to the reflectarray diameter (D), which is known as blockage ratio [5]. It should be noted here that blockage diameter is not the physical diameter of the feed antenna, but an effective projected diameter that is related to the physical volume of the feed. The undesirable effects of aperture blockage are primarily an increase in side‐lobe level and a reduction in antenna gain, which further degrade as this ratio is increased. Similar to reflectors, as long as the blockage ratio is below 0.2, the effects of blockage will not be significant [5], however, this is more challenging when one deals with reflectarrays with smaller aperture sizes. The problems associated with aperture blockage can be avoided by offsetting the feed antenna, which is schematically shown in Figure 3.9(b). The basic characteristics of the offset system are similar to the symmetric case, and as such the offset system is generally preferred, especially for moderate size reflectarrays with nominal gain of

D0

D

D

Projected Aperture

(a)

(b)

Figure 3.9  Aperture blockage in front‐fed reflectarray systems: (a) symmetric, (b) offset.

System Design and Aperture Efficiency Analysis

about 30 dB. The offset system, however, exhibits a higher cross‐polarization level due to the fact that the focal‐length‐to‐diameter for offset systems is usually smaller [3]. In addition, for the same physical aperture diameter (D), the gain of the offset system is lower due to the smaller projected aperture effect. Nonetheless, elimination of feed blockage with the offset system is a significant advantage over the symmetric configuration. As discussed earlier, efficiency analysis is a critical step in the system design. In this section, we will present some numerical studies on the aperture efficiency for offset reflectarray systems, which would give the reader a better understanding on the importance of efficiency analysis in system design. The basic formulations for calculating the aperture efficiency of reflectarray systems were outlined in the previous section and it was shown that the aperture efficiency of a reflectarray system is a function of the feed (position, orientation, and radiation pattern), reflectarray element pattern, and shape of the aperture. As discussed earlier, circular apertures are generally the optimum choice for reflectarrays, and thus selected for this study. A reflectarray with a diameter of 20λ is considered for the design here. In general, the element pattern shape has very little effect on the efficiency of the system [6]. Moreover, while the reflection characteristics of different element shapes are quite different, in most cases the element pattern is quite similar, as seen in Chapter 2. Thus, here we consider a cos(θ) pattern model for the element. Now let’s study the aperture efficiency of an offset system configuration. For offset configurations, the offset angle is typically selected to minimize the feed blockage. Here we select a feed pattern with q = 6.5, and an offset angle of 20°, where the feed points to the geometrical center of the aperture. The aperture efficiency as a function of feed distance to reflectarray aperture is given in Figure 3.10(a), where it can be seen that the general trend of the efficiency factors is similar to the symmetric case. The optimum position of feed is at a distance of 15.2λ from the reflectarray aperture, yielding an aperture efficiency of 75.69%. It can be seen that with optimum feed placement, the offset system efficiency can achieve an aperture efficiency close to the symmetric case (77.34%), with the added advantage of minimized feed blockage. It is important to note that the offset angle plays a pivotal role here. For the same feed pattern, the aperture efficiency as a function of both feed position and offset angle are given in Figure 3.10(b). It can be seen that for small values of offset angle (less than 20°), a high aperture efficiency can be achieved, however, as the offset angle increases beyond this, the system can no longer yield high efficiency. In addition, for reflectarrays, a large offset angle for the feed will result in elements being excited with very oblique angles, which will further degrade the performance of the system. As such, in general it is desirable to keep the offset angle to a minimum. In the studies presented so far, the feed antenna was pointed to the geometrical center of the reflectarray aperture. While this is the optimum choice for symmetric designs, for offset systems the optimum beam pointing direction of the feed is not the geometrical center of the array. To study this, here we consider the same offset reflectarray system where the feed is placed with an offset angle of 20°, at a height of 15.2λ from the aperture. The beam pointing direction is then adjusted by displacing the feed along the x‐axis in Figure 3.1. The aperture efficiency as a function of beam pointing position (X0) of the feed is given in Figure 3.11. It can be seen that for offset systems, the optimum beam point for the feed is slightly below the geometrical center of the aperture. In this case, this point is −0.6λ, and with this setup the system will achieve an aperture

61

Reflectarray Antennas

100

Efficiency (%)

90 80 70 ηi ηs

60

ηa

50 10

15

20

25

z (λ)

(a) 40

75 70

θi (deg.)

30

65 60

20

55 50

10 45 40

0 10

15

20

25

z (λ)

(b)

Figure 3.10  Efficiency for an offset reflectarray with q = 6.5: (a) efficiency as a function of feed position for θoffset = 20°, (b) aperture efficiency as a function of feed position and θoffset. Figure 3.11  Efficiency for an offset reflectarray as a function of feed beam pointing position on the aperture.

100 80

Efficiency (%)

62

60 40

ηi ηs

20

ηa 0 –10

–5

0 x0 (λ)

5

10

System Design and Aperture Efficiency Analysis 1

10

5

0

0.5

y (λ)

5

y (λ)

1

10

0

0.5

–5

–5 –10 –10

0

10

–10 –10

0

x (λ)

(a)

0

10

0

x (λ)

(b)

Figure 3.12  Amplitude distribution on the reflectarray aperture: (a) feed points to the geometrical center, (b) feed points to the optimum point.

efficiency of 76.1%. The amplitude distribution on the reflectarray when the feed points to the geometric center, and the optimum position are given in Figure 3.12(a) and (b), respectively. Note that with these optimized values (the offset angle, feed location, and beam pointing direction), the efficiency of the offset reflectarray is only 1% lower than the symmetric case. 3.3.2  Edge Taper and Edge Diffraction An important consideration in designing a reflectarray system is the relative field strength at the edge of the reflectarray aperture with respect to the maximum, known as edge taper (ET). Similar to reflector antennas, the fields at the edges of the reflectarray aperture will inevitably exhibit diffraction that causes the rays to diverge. These edge diffracted fields degrade the radiation characteristics of the antenna since they no longer exhibit the ideal properties of a collimated wave. To minimize the effects of edge diffraction on the radiation performance of the antenna, the illumination level at the edges, that is, the ET, should be low. For front‐fed symmetric reflectors, ET is defined as the ratio of field intensity at the reflector rim to the intensity at its center and is typically given in decibels. The same formulation is adopted here for reflectarrays. The geometrical setup of a front‐ fed symmetric reflectarray is given in Figure 3.13. The two parameters that define this reflectarray system are the aperture diameter (DRA) and the distance from the focal point to the aperture surface (HF), which is similar to parabolic reflectors, where the diameter and focal length fully describe the shape of the parabola. The analogy between parabolic reflectors and reflectarrays will be discussed later on in Section 3.4. For the reflectarray setup here, the half subtended angle is given by



0

tan

1

DRA . (3.21) 2HF

63

64

Reflectarray Antennas

Figure 3.13  Geometrical setup of a front‐fed symmetric reflectarray.

x z

RE Focal Point Center (C) θ0

RC

DRA

Reflectarray

HF

Within this subtended angle, the illumination varies from a maximum value at the center, to a minimum value at the edge. As before, we use the cosq pattern model for a balanced feed, where the illumination at any point on the aperture can then be given by



I

cos q

f

rf

. (3.22)

Here rf is the distance from the focal point to any point on the aperture, and θf is the angle between rf and the surface normal (Rc). The illumination at the center of the reflectarray aperture can then be given as



I RAC

cos q

0o

1 . (3.23) HF

HF

Similarly, the illumination at the edge of the reflector is given by

I RAE

cos q

0

RE

, where

RE

DRA . (3.24) 2 sin 0

It should be noted here that, in this setup it is assumed that the system is rotationally symmetric. In other words, the reflectarray has a circular aperture, which as demonstrated earlier provides the highest aperture efficiency, and the feed points to the geometrical center. For the front‐fed symmetric reflectarray system, the ET is then given as



ETRA

20 log10

I RAE I RAC

20 log10 2

HF sin DRA

0

cos q

0

. (3.25)

System Design and Aperture Efficiency Analysis

It is worthwhile pointing out that this formula also allows one to select the desired feed power pattern, the value of q in this case, based on the desired value of ET. For reflectarrays this is given by ETRA

q



DRA 10 20 2 H F sin 0 log cos 0

. (3.26)

As a simple example here, we study a reflectarray with a circular aperture diameter of 20λ. In the first case, we fix the geometrical setup of the system, that is, HF/DRA = 0.8, and study ET as a function of feed pattern. This is given in Figure 3.14(a), where it can be seen that a monotonous decrease in ET is observed as the value of q increases. The typical or commonly used value for ET is about −10 dB, which for this system corresponds to a q value of 5.98. The illumination on the aperture for this q value is also given 0

Edge Taper (dB)

–5 –10 –15 –20 –25

(a)

5

10

15

q 0

10

y (λ)

5

–5

0

–5

–10 –10

(b)

0 x (λ)

10

–10

Figure 3.14  Aperture illumination and edge taper for a front‐fed symmetric reflectarray: (a) ET versus q for HF/D = 0.8, (b) normalized illumination on the aperture for q = 5.98 in dB scale.

65

Reflectarray Antennas

in Figure  3.14(b). It is important to note that in this case, a feed antenna has to be designed that satisfies this pattern requirement. For the second case, we study ET as a function of feed position (HF) which is the practical approach when an available feed antenna has to be used for the system. We select the same aperture size of 20λ, and a feed antenna with a q value of 7.4, which corresponds to a directivity of 15 dB. Here we study HF/DRA in the range from 0.4 to 1. The results are given in Figure  3.15, where it can be seen that the ET changes quite rapidly for smaller values of HF/DRA. With this feed, for a 10 dB ET, the value of HF/DRA is 0.9. It is worthwhile to note that while the typical value of ET is about −10 dB, slightly lower value of edge illumination can yield better aperture efficiency. Now let us study the offset feed configuration. The geometrical setup of an offset fed reflectarray is shown in Figure  3.16. As with reflectors [5], unfortunately the definition of ET that was introduced earlier for symmetric systems is ambiguous for offset configurations, since the taper on the reflectarray edge will be a function of φa. One approach to quantify the ET is to study it on the upper, lower, and side edges.

–5

Edge Taper (dB)

–10 –15 –20 –25 –30 –35 0.4

0.6

0.8

1

HF/DRA

(a) 10

0

5

y (λ)

66

0

–5

–5

–10 –10

(b)

0

10

–10

x (λ)

Figure 3.15  Aperture illumination and edge taper for a front‐fed symmetric reflectarray: (a) ET versus HF for a fixed feed pattern (q = 7.4), (b) normalized illumination on the aperture for HF/DRA = 0.9 in dB scale.

System Design and Aperture Efficiency Analysis

Figure 3.16  Geometrical setup of an offset reflectarray system.

Upper (U)

U

φa

C Side (S)

DRA/2 x

RUE z

C x0

L Focal Point

θoffset

B

Reflectarray

RFP

Lower (L)

RLE

HF

Note that unlike the symmetric case, where the feed antenna points to the geometrical center of the circular aperture, the offset case may have its feed pointed to a different point as shown in Figure 3.16. In any case, the system can be properly defined by a unique set of parameters, namely DRA , HF, ΔB, and θoffset which are shown in Figure 3.20. Similar to the symmetric case, the radial distance from the focal point is computed for several points on the aperture. Note that in the offset configuration, aperture efficiency can be maximized when the feed is pointing to a point slightly below the center [6], shown as B in Figure 3.16, therefore this geometrical setup is adopted with the equations given here. RU

RFP

RC DRA 2

cos

HF offset

2 B

, focal point to B, (3.27)

2 RFP 2 RFP

2

B cos

2 RFP 2 RFP

B

offset

DRA 2

2

B

, focal point to C , (3.28)

cos

offset

2

, focal point to upper edge, (3.29)

RL

DRA 2

2

2 RFP 2 RFP

B

DRA 2

B

cos

2

offset

, focal point to lower edge, (3.30)



RS

DRA 2

2

RC2 , focal point to side edges. (3.31)

67

68

Reflectarray Antennas

Figure 3.17  The side‐view of an offset reflectarray system showing the edge and feed subtended angles.

θUE

RUE DRA

C Focal Point

θoffset

B

θU

RFP Reflectarray

θL RLE θLE HF

In order to compute the illumination at the edge, it is also necessary to compute the angle at each point with respect to the feed focal point as shown in Figure 3.17. These angles are given by



UE

cos

1

LE

cos

1

S

tan

1

HF , RU

U

UE

HF , RL

L

LE

offset , for upper edge, (3.32)

offset , for lower edge, (3.33)

DRA , for side edges. (3.34) 2 RC

Note that for the offset configuration the subtended feed angle is θL + θU. The illumination at any of the desired edge points can now be computed using (3.13). As with the symmetric case, taper has to be computed with respect to the maximum illumination, however, in the offset configuration the position of maximum illumination is unknown, and must be determined. With the geometrical setup of Figure 3.16, the maximum illumination is along the x‐axis, and typically at ‐x0 > xmax > ‐D/2. In any case once the maximum illumination is computed, ET can easily be determined at the desired points on the edge of the reflectarray aperture using





ETU

20 log10

ETL

20 log10

IUE

I MAX I LE

I MAX

20 log10

cos q U I MAX RU

, for upper edge, (3.35)

20 log10

cos q L I MAX RL

, for lower edge, (3.36)

System Design and Aperture Efficiency Analysis 0 Normalized Illumination (dB)

Figure 3.18  Illumination on the offset reflectarray aperture: (a) along x‐axis, (b) along y‐axis.

B

C

–5

–10

–15 –10

–5

(a)

0 x-axis (λ)

5

10

5

10

Normalized Illumination (dB)

0 C –5

–10

–15 –10

(b)



ETS

20 log10

I SE

I MAX

20 log10

cos q S I MAX RS

–5

0 y-axis (λ)

, for side edges. (3.37)

As an example, here we study an offset reflectarray system with a circular aperture diameter of 20λ, HF = 16λ, θoffset = 20°, and x0 = λ. The feed antenna has a q value of 7.4, which corresponds to a directivity of 15dB. The illumination along the x‐ and y‐axes of the reflectarray aperture is given in Figure 3.18. For this system, the maximum illumination occurs at x = −1.8515λ. As discussed earlier, in all cases the maximum illumination is at a point that is slightly lower than where the feed points to, that is, point B in Figure 3.16. The aperture illumination is also given in Figure 3.19. As discussed earlier, for an offset system ET is ambiguous. However, the formulation outlined in this section allows one to observe the taper on the aperture at certain edge points, and control the ET by proper feed design or placement. In any case optimum system design is generally conducted by means of an efficiency analysis as demonstrated earlier.

69

Reflectarray Antennas 10

0

5

y (λ)

–5 0

–5 –10 –10 –10

0

(a)

10

x (λ)

–8

Edge Taper (dB)

70

–9 –10 –11 –12 –13

(b)

0

50

100 ϕaperture (degrees)

150

Figure 3.19  (a) Aperture illumination of an offset reflectarray. (b) ET as a function of polar angle on the reflectarray aperture.

3.4 ­The Analogy between a Reflectarray and a Parabolic Reflector Parabolic reflector antennas have long served as the most suitable antenna for high‐ gain operation in both terrestrial and spaceborne applications [1], [2]. Reflectarray antennas are essentially built based on the concept of reflector antennas. Given the fact that reflector antenna technology is mature and well developed, it certainly is beneficial to take advantage of the developments that have been made in reflector design over generations. To this end one should be able to describe a reflectarray system that exactly mimics the parabolic reflector system. However, in many cases, the design parameters for the reflectarray don’t correlate directly to the reflector design parameters. This is due to the fact that for each antenna, the design parameters are given in manner that best describes its own system. While this is less of an issue for axisymmetric designs, the offset configuration is more complex. In particular,

System Design and Aperture Efficiency Analysis

many important reflector design parameters such as focal depth are typically misinterpreted, while other parameters such as feed tilt angle, and offset height are essentially nonexistent in offset reflectarrays [7]. In this section, we will first review the offset parabolic reflector design characteristics, and then define the analogous relationship between offset reflector and reflectarray systems. We then present the mathematical formulation that converts the design parameters of the parabolic reflector system to the reflectarray design parameters and vice versa. These equations simplify the configuration setup in offset reflectarrays, by using a coordinate system and a set of parameters that is well suited for design and analysis of offset reflectarrays, and can be advantageous for complex reflectarray configurations, such as offset dual‐reflectarray systems [8], [9]. 3.4.1  The Offset System Configurations In an offset reflector configuration, many parameters exist that can be utilized to achieve the required design goals [5]. However, the most important reflector design parameters, which can fully describe a focused offset system, can be given in terms of four key parameters. The geometrical model of the offset parabolic reflector system is given in Figure 3.20(a), where the design parameters are: D = reflector diameter, F = focal length, H = offset height, ψB = feed tilt angle.

●● ●● ●● ●●

Note that the first three parameters completely describe the geometrical setup of the parabolic reflector; however, since in general the primary feed does not point to the reflector center (C), for a full system specification it is necessary to specify the feed tilt

x

D

C

Reflector Center

x′

z

B

DRA C′ B′

Reflectarray Center z′ ∆B θoffset

Focal Point

Focal Point

ψB

H

HF

F

(a)

(b)

Figure 3.20  The geometrical models and key parameters of the offset systems: (a) parabolic reflector, (b) reflectarray.

71

72

Reflectarray Antennas

angle. In the general case, the feed points to point B on the reflector surface, as shown in Figure 3.20(a). As discussed in earlier sections, the offset reflectarray system can also be fully specified with four key parameters. The geometrical model of the offset reflectarray system is given in Figure 3.20(b), where the design parameters are: ●● ●● ●● ●●

DRA = reflectarray diameter, HF = focal point to aperture distance, ΔB = offset distance from reflectarray center, θoffset = feed offset angle.

Similar to the reflector system, the primary feed points to a point on the reflectarray aperture (B′) which is not necessarily the geometrical center, where the optimal position for B′ is determined by efficiency analysis. Note that in order to distinguish the design parameters of the two systems, we use the non‐prime coordinates for the reflector system, and the prime coordinates for the reflectarray system. The two sets of parameters described here for these offset configurations make it possible to fully describe the antenna systems. In order to establish an analogy between these two systems, one must be able to derive the relationship between the parameters in the two sets. It should be noted here that in the reflector system, the main beam of the antenna points in the z‐direction in the coordinate setup of Figure 3.20(a). For the reflectarray system, however, one has the option to control the beam direction by adding a progressive phase to the aperture. More discussion on this will be given in the following sections. 3.4.2  Analogous Offset Reflector The key system design parameters for offset reflectors and reflectarrays were described in the previous sections. In general, in order to analyze the antennas one makes use of a coordinate system that best suits the geometrical setup, which are different for the reflector and reflectarray systems. In order to establish the analogy between these two systems, we will use the coordinate setup of the reflector antenna. It is implicit that once the relationship between the design parameters is determined, one can describe the systems in any desired coordinate system. 3.4.2.1  Transformation from Reflector to Reflectarray System

The geometrical models of the offset configurations in the reflector coordinate setup are shown in Figure 3.21. Here we consider the case where the reflector design parameters (F, D, H, ψB) are given. First, we derive some additional parameters for the reflector using the four main parameters that would aid in our derivations. The angle subtended to the reflector lower tip, and the distance from focal point to this point are given as

L

RL

2 tan

1

2F 1 cos

H , (3.38) 2F L

. (3.39)

System Design and Aperture Efficiency Analysis

Figure 3.21  The analogous geometrical setup of the offset reflector and reflectarray systems.

Reflector

D

B

Reflectarray B′

RU

ψU H

ψL

Focal Point

RL F

Similarly, the angle subtended to the reflector upper tip, and the distance from focal point to this point are given by

U

RU

2 tan

1

2F 1 cos

D H , (3.40) 2F U

. (3.41)

Note that with this setup, the feed subtended angle, ψS, is identical in both systems and is given by

S

L . (3.42)

U

With these parameters defined, now let us derive the reflectarray system parameters in this coordinate setup. The system setup is given in Figure 3.22, where the feed subtended angle is S U L. These angles can be given in terms of reflector parameters as

L



DRA

L , and U

B

U

B . (3.43)

The diameter of the reflectarray antenna can now be calculated in terms of these parameters as RU2

RL2 2 RU RL cos

S

. (3.44)

As discussed earlier, with the reflectarray antenna one can direct the main beam to any desired direction. However, since here we aim at describing analogous systems, the main beam for the reflectarray points in the same direction as the reflector, as shown in Figure 3.23. Note that in the current setup, the reflectarray aperture is tilted by a certain angle with respect to the reflector coordinates which is given by

tilt

cos

1

D . (3.45) DRA

73

74

Reflectarray Antennas

Figure 3.22  The geometrical setup of the offset reflectarray in the reflector coordinates.

DRA

C′ B′

∆B

RU θoffset θU Focal Point

θL

RL

Figure 3.23  The main beam direction and geometrical setup of the offset reflectarray system.

DRA

D

Main beam

B′ θBRA

θtilt

θoffset Focal Point

ψB

To direct the reflectarray main beam in the same direction as the reflector, one must have θBRA = θtilt, where θBRA is the angle with respect to the normal on the reflectarray aperture. The feed offset angle for the reflectarray can then be given as

offset

BRA . (3.46)

B

The distance from focal point to B′ is given as RFP

RL

sin

L

2

sin

2

offset

. (3.47)

offset

The focal point to reflectarray aperture distance, and the feed point offset distance from reflectarray center can now be given in terms of these parameters as

System Design and Aperture Efficiency Analysis



HF B

RFP cos 1 2

2 RFP

offset

, (3.48)

RU2

2 RFP RU cos

2 RFP

U

RL2 2 RFP RL cos

L

. (3.49)

The equations given in this section can fully describe the analogous reflectarray system in terms of the reflector design parameters. In particular, the four parameters of the offset reflectarray are computed from equations (3.44), (3.46), (3.48), and (3.49), respectively. 3.4.2.2  Transformation from Reflectarray to Reflector System

Now let’s consider the case where the reflectarray design parameters (HF, DRA, ΔB, θoffset) are given. The geometrical setup of the reflectarray is essentially identical to the previous case shown in Figure 3.22. Similarly, we derive the distance from focal point to the reflectarray upper and lower tips using





RL

2 RFP

DRA 2

RU

2 RFP

DRA 2

2 B

2 RFP

DRA 2

B

cos

2 RFP

DRA 2

B

cos

2 B

2

offset

, (3.50)

2

offset

, (3.51)

where RFP is the distance from focal point to B′ and can be derived using (3.47). The feed angles subtended to the upper and lower tips are then given as





L

U

cos

cos

DRA 2

1

2 B

(3.52)

2 RFP RL 2

DRA 2

1

2 RL2 RFP ,

B

RU2

2 RFP

.



(3.53)

2 RFP RU

Using the geometrical formula for a paraboloid, the distance from focal point to the upper and lower edges can also be given using

RL RU

2F 1 cos 1 cos

, (3.54)

L

2F L

S

, (3.55)

where ψS is the total feed subtended angle as given in (3.41). The angle subtended to the reflector lower tip can be derived by solving these two equations simultaneously as

L



Re

i ln

RL RU

S

2i sin

RL RU e

i

2 S

.



(3.56)

75

76

Reflectarray Antennas

The reflector design parameters can now be computed using F

RL 1 cos 2



H

RL sin

L



D

RU sin

L





B

cos

1

L

, (3.57)

, (3.58) S

D DRA

H , (3.59) offset . (3.60)

These equations make it possible to describe an analogous offset reflector system in terms of the reflectarray design parameters. One important note here is that in the reflectarray to reflector transform, the direction of the main beam of the reflectarray cannot be defined independently. In the setup outlined here, the direction of the reflectarray main beam is in the same direction as the reflector. However, as discussed earlier, the reflectarray can be designed to scan the beam in any desired direction, but in any case, all the important reflector characteristics such as focal depth, offset height, and feed tilt angle, will be identical. In other words, with the same configuration setup for both antennas (Figure 3.21), the reflectarray allows for more flexibility in the system design. 3.4.3  Example of Analogous Offset Systems The formulations presented in the previous section allow one to accurately determine the important parameters of offset reflectarray systems [10], [11], in a fashion that has been well established for parabolic reflector antennas. Here, let us study the performance of an analogous parabolic reflector and reflectarray system. A Ka‐ band reflectarray antenna with an aperture diameter of 190 mm, designed for the center frequency of 32 GHz, is considered. We select a balanced feed antenna with a q value (cosqθf radiation pattern model) of 6.5. The optimal values of the reflectarray design parameters are determined based on efficiency analysis, and are summarized in Table 3.1. For the analogous reflector antenna, the design parameters are determined using the formulation presented in Section 3.4.2, and are also summarized in this table. The F/D ratio for this system is 0.83. As discussed earlier, the main beam direction of the reflectarray (θBRA) is determined based on the parabolic reflector parameters, and should be equal to 19°. The 2D‐model (cross‐sectional view) of this parabolic reflector Table 3.1  Design parameters for the offset reflectarray and reflector. Reflectarray Reflector

DRA

HF

ΔB

θoffset

190 mm

144.71 mm

6 mm

20°

D

F

H

ψB

179.64 mm

148.93 mm

12.78 mm

39°

System Design and Aperture Efficiency Analysis

y′-axis (mm)

z

200

0

100

–50

y

50 –50 0 x′-axis (mm)

x

(a)

300

50

0

(b)

Figure 3.24  (a) The parabolic reflector antenna model in FEKO. (b) The element phase distribution on the analogous reflectarray antenna aperture.

and reflectarray system is similar to that shown in Figure 3.21. For the parabolic reflector antenna, the design is essentially complete, however, for the reflectarray antenna, the array nature of the system requires one to also specify the element spacing. Here we set the element spacing to 4.7 mm (~λ/2 at 32 GHz). This corresponds to 1184 elements on the aperture of the reflectarray. In this study, we are comparing the idealized performance of both systems, thus for the reflectarray we use ideal elements, that is, elements with no reflection loss (|Г| = 1), and the exact (non‐quantized value) required phase shift. Also for the parabolic reflector we use perfect electric conductor. The 3D model of the parabolic reflector antenna in FEKO [12], and the required element phase shift on the aperture of the reflectarray antenna are given in Figure 3.24. It is important to note that the radiation patterns of these two antennas are almost identical, as expected, and the interested readers are referred to [13].

­References 1 C. J. Sletten, Reflector and Lens Antennas: Analysis and Design Using Personal Computers,

Artech House Inc., 1988.

2 M. E. Cooley, and D. Davis, “Reflector antennas,” in Radar Handbook, 3rd Edn, McGraw‐

Hill, 2008.

3 W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, 3rd Edn, John Wiley &

Sons Inc., 2012.

4 C. A. Balanis, Antenna Theory: Analysis and Design, 3rd Edn, John Wiley & Sons Inc., 2005. Y. Rahmat‐Samii, “Reflector Antennas,” in Antenna Handbook: Theory, Applications, and 5

Design, Y. T. Lo and S. W. Lee, Van Nostrand Reinhold, 1988.

6 A. Yu, F. Yang, A. Z. Elsherbeni, J. Huang, and Y. Rahmat‐Samii, “Aperture efficiency

analysis of reflectarray antennas,” Microwave and Optical Technology Letters, Vol. 52, No. 2, pp. 364–372, Feb. 2010.

77

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Reflectarray Antennas

7 P. Nayeri, A. Z. Elsherbeni, and F. Yang, “The analogy between offset configurations of

8

9 10 11 12 13

parabolic reflectors and reflectarrays,” IEEE Antennas and Propagation Society International Symposium, Vancouver, Canada, July 2015. E. Almajali, D. McNamara, J. Shaker, and M. R. Chaharmir, “Derivation and validation of the basic design equations for symmetric sub‐reflectarrays,” IEEE Trans. Antennas Propag., Vol. 60, No. 5, pp. 2336–2346, May 2012. C. Tienda, M. Arrebola, J. A. Encinar, G. Toso “Analysis of a dual‐reflectarray antenna,” IET Microw. Antennas Propag., Vol. 5, No. 13, pp. 1636–1645, 2011. J. Huang and J. A. Encinar, Reflectarray Antennas. New York, NY, USA: Wiley‐IEEE, 2008. D. M. Pozar, S. D. Targonski, and H. D. Syrigos, “Design of millimeter wave microstrip reflectarrays,” IEEE Trans. Antennas Propag., Vol. 45, No. 2, pp. 287–296, Feb. 1997. FEKO v7.0, EM Software & Systems Inc., 2014. P. Nayeri, A. Z. Elsherbeni, and F. Yang, “The analogy between offset configurations of parabolic reflectors and reflectarrays,” IEEE Antennas and Propagation Society International Symposium, Vancouver, Canada, July 2015.

79

4 Radiation Analysis Techniques The design and analysis procedure for reflectarray elements and the reflectarray system were outlined in detail in Chapters 2 and 3, respectively. The next task is to calculate the radiation characteristics of reflectarray antennas, such radiation pattern, directivity, gain, and so on. This chapter focuses on radiation analysis techniques for reflectarray antennas. As discussed in earlier chapters, reflectarrays combine the  numerous advantages of both printed arrays and parabolic reflectors and create the new hybrid high‐gain antenna [1]. In terms of analysis approaches for reflectarrays, it  is  implicit that they are essentially adapted from the analysis techniques developed for their ­predecessors over generations. Various analysis approaches have been developed to calculate the radiation characteristics of the reflectarray. Most reflectarray analysis approaches approximate the elements on the reflectarray aperture as identical elements that form an array, and then array summation or far‐field transformation of currents is used to calculate the radiation pattern of the reflectarray antenna [2], [3]. Accurate characterization of the reflection coefficients of the reflectarray elements holds a significant importance in the radiation analysis for these approaches. These coefficients are therefore usually obtained with a full‐wave simulation technique, using an infinite‐array approach that takes into account the mutual coupling between the elements as described in Chapter 2. Alternatively, the current distributions on the reflectarray elements (obtained from the full‐wave simulations under the local‐periodicity approximation) could be directly used, and the radiation pattern would then be calculated using far‐field transformation of the currents for the total array [4]. The great advantage of these analysis methods is the fast computational time. However, it is implicit that due to the approximations in the analysis, some discrepancies with practical results may be observed. On the other hand, a full‐wave simulation of the entire reflectarray can provide accurate results. The electrically large size of the reflectarray antenna aperture, combined with hundreds of elements with dimensions smaller than a wavelength, demands a significantly high computational time and large resources for full‐wave analysis [5], [6]. While in many cases a full‐wave simulation will be advantageous at the final stage of a design, fast computational approaches are still necessary tools for the initial design. This is in addition to several stages of optimization that may be required for a reflectarray antenna design [7], [8]. In this chapter, we first present two basic methods for analysis of the radiation performance of reflectarray antennas. The two different approaches, namely the array theory Reflectarray Antennas: Theory, Designs, and Applications, First Edition. Payam Nayeri, Fan Yang, and Atef Z. Elsherbeni. © 2018 John Wiley & Sons Ltd. Published 2018 by John Wiley & Sons Ltd.

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Reflectarray Antennas

method and the aperture field method, are described, and numerical results are then presented for various reflectarray configurations. The advantages and limitations for each method are discussed. Moreover, some important radiation analysis concepts such as directivity, gain, and loss budget calculations are described in detail. Next, full‐ wave simulation approaches are discussed, and comparison between the theoretical analysis techniques and full‐wave simulations are also presented, which shows a close agreement.

4.1 ­Array Theory Approach: The Robust Analysis Technique Analysis of electrically large antennas is a topic of great interest in antenna engineering, and in the broad sense it can be divided into analog and digital techniques. Continuous large antennas, such as the parabolic reflector, form the analog section and must be analyzed using integrals that are usually time consuming. Arrays on the other hand form the digital section and thus can be analyzed using fast and simple summations. The reflectarray antenna by nature possesses the basic properties of antenna arrays, and as such one should be able to use the array theory technique for analysis. However, given that this antenna is space‐fed (in contrast to conventional arrays which have a feed network), the array theory analysis for reflectarray antennas demands some further considerations, which will be outlined in this section. 4.1.1  Idealized Feed and Element Patterns In the array theory approach, the most primary step is to determine the element tapers. For the reflectarray antenna, the space‐fed nature of the system implies that the taper is directly dependent on the position and radiation pattern of the feed antenna, and as such characterization of the feed is of paramount importance. The feed radiation pattern model was described in Chapter 3 and can be applied in the array theory analysis; however, polarization of the feed is not considered in the analysis here. We will provide more details about feed polarization later on in this chapter. For simplicity, the radiation pattern of the feed antenna is assumed to be rotationally symmetric and is given by EF

f ,

f

cos q 0

f

for 0 f elsewhere

/2

, (4.1)

where similar to (3.10), θf is the feed elevation angle and q is the feed power pattern. The simplified feed radiation pattern model described here makes it very easy to compute the illumination of each element on the reflectarray aperture. However, note that for the reflectarray antenna, accurate modeling of the array excitation also requires one to include the reflection coefficients of the element when computing the element taper. More discussion on this is given in the next section. For an antenna array, the radiation pattern of the antenna is essentially the summation of the radiation pattern of all of its elements. As such another important step in the array theory analysis for reflectarrays is to characterize the radiation pattern of the ­elements. Here we use the same element pattern described in Section  3.2.2 for the

Radiation Analysis Techniques

element, that is, a rotationally symmetric pattern modeled with a cosine q function, which is given by



EE

p,

cos qe 0

p

for 0 p elsewhere

p

/2

, (4.2)

where θp is the elevation angle in the local coordinate system for any element on the reflectarray aperture as shown in Figure 3.4. 4.1.2  Element Excitations and Reflectarray Radiation Pattern The conventional array theory can be implemented to calculate the far‐field radiation pattern of the reflectarray antenna. The radiation pattern of a 2D planar array with M × N elements can be calculated as  E ( uˆ ) =

M N



 

∑∑Amn ( uˆ ) ⋅ I ( rmn ) ,

m =1 n =1

uˆ = xˆ sin θ cosϕ + yˆ sin θ sin ϕ + zˆ cosθ ,

(4.3)

where A is the element pattern vector function, I is the element excitation vector func tion, and rmn is the position vector of the mnth element. The coordinate system for the reflectarray antenna is given in Figure 4.1. To simplify calculations, usually one uses scalar functions in the analysis. For the element pattern function, A, a scalar approximation considers a cosine qe model for each element with no azimuth dependence, that is, 



Amn (θ ,ϕ ) ≈ cos qe (θ ) ⋅ e jk ( rmn ⋅u ) . (4.4) ˆ

Z uˆ

rf

Observation direction uˆ o

Main beam direction

Y

rmn (m,n)th element

X

Figure 4.1  Coordinate system of the reflectarray antenna for array theory analysis.

81

82

Reflectarray Antennas

The element excitation function I(m,n), is determined by the incident field and e­ lement property. By approximating the feed horn pattern function using a cosine q model and taking into account the Euclidian distance between the feed horn and the element, the illumination of the aperture can be obtained. The element excitation can then be expressed as I m,n

Cos q f f m,n e   rmn r f

  jk rmn r f

mn

ej

mn

. (4.5)

 Here, θf is the spherical angle in the feed coordinate system and r f is the position vector of the feed. In addition, for each element this excitation can take into account the receive mode pattern, that is |Γmn|. This pattern is also modeled by a cosine qe function based on the local element coordinates, that is, Cos qe e m,n . (4.6) mn It should be noted that this definition of |Γmn| is for the receive/transmit (R/T) model. In the scattering model analysis approach, this value is obtained directly from the unit‐ cell analysis. The required phase delay of the mnth element, ϕmn, is designed to set the main beam in the ûo direction as described in Chapter 2. It should be noted that since the infinite‐ array approach is often used for analysis of reflectarray elements, ϕmn does contain the mutual coupling effects under local periodicity approximation. With these approximations, the radiation pattern can be simplified to the scalar form M N



E (θ ,ϕ ) = ∑∑Cos qe θ m=1n=1

Cos q f θ f ( m,n ) − jk ( rmn −rf −rmn ⋅uˆ ) e Cos qe θ e ( m,n )e jφmn . (4.7)   rmn − r f

The radiation pattern calculation method described here uses a conventional array summation technique. In general, the array theory formulation will yield good main beamwidth, beam direction, and general pattern shape; however, since the polarization of the feed horn and elements are not accounted for in the simplified cosine q model, the cross‐polarization of the reflectarray antenna is not calculated in this procedure. In summary, the advantages and disadvantages of this approach are: Advantages: Simplicity of the formulation and program development ●● Fast computational time Limitations: ●● Cross‐polarization is not modeled ●●

4.2 ­Aperture Field Approach: The Classical Analysis Technique 4.2.1  Complex Feed Patterns Antenna feeds play a major role in the radiation performance and efficiency of the reflectarray system. The balanced feed cosq pattern described in Chapter 3 is a popular model for the feed antenna pattern, however, more accurate analysis may require one to

Radiation Analysis Techniques

use the radiation pattern of the practical feed antenna. A variety of antenna configurations such as open‐ended waveguides (OEWG), microstrip patch arrays, pyramidal and conical horns, hybrid‐mode horns, and corrugated horns have been used for reflector feeds: see [9], [10], and [11]. Analytical solutions for the far‐field radiation pattern of these feed antennas are available in literature [12] and can be used directly for the radiation analysis of the reflectarray. In this section, we will briefly review the analytical radiation pattern of corrugated circular aperture or dual mode horns. A circularly symmetric feed radiation pattern typically provides the best aperture efficiency for reflectors. Practical realization of such pattern can be accomplished by corrugated or dual mode horn antennas. The pattern of these antennas is similar to the H‐plane pattern of a TE11 mode radiation from an open‐ended circular waveguide, which is mathematically given as E

f

ka

2

ka

1.841 2

2

1.841

ka sin 2

ka

f

.

J 0 ka sin 1



f

J 2 ka sin

ka sin f 1.841

2

f

. (4.8)

Here ka is the electrical circumference of the waveguide. With a fixed phase center, the complete radiation pattern model for the feed is then given by



e − jkrf E F ( r f ,θ f ,ϕ f ) = A0 θˆ E (θ f ) cosϕ f − ϕˆ E (θ f ) sin ϕ f  , for x polarized,   rf



e − jkrf E F ( r f ,θ f ,ϕ f ) = A0 θˆE (θ f ) sin ϕ f + ϕˆ E (θ f ) cosϕ f    r f , for y polarized.

(4.9)

(4.10)

where rf is the vector from the feed phase center to any point on the reflectarray aperture. 4.2.2  Field Transformations from Feed to Aperture and Equivalent Surface Current The next step in the analysis is to determine the field radiated by the feed at each reflectarray element in terms of the Cartesian components in the reflectarray coordinate system [13]. A flowchart of this procedure is given in Figure 4.2. First, the incident field is calculated at the center point of each reflectarray element using the appropriate feed pattern model. Then the field is transformed into Cartesian components using the following equation E xF E yF

EzF

sin f cos sin f sin cos f

f f

cos f cos cos f sin sin f

f f

sin cos 0

f f

0

E Ff . (4.11) E Ff

In this equation (θf, φf) are the spherical angles of the reflectarray elements in the feed coordinate system. Since the spherical angles change with position of the element, this

83

84

Reflectarray Antennas For each element of the reflectarray (m = 1:Nx, n = 1:Ny) E F(m,n)(θf,φf)

E F(m,n)(Xf,Yf,Zf)

Using 4.11

Rotate coordinates about (X, Y, Z) by (α, β, γ) depending on the system Function rotx(array, α) Function roty(array, β) Function rotz(array, γ)

E inc (X, Y, Z) (m,n)

Figure 4.2  Flowchart of the transformations from feed to array coordinates.

transformation is carried out for each element of the reflectarray. It is worth emphasizing the superscript “F,” which indicates the fields here are in the Cartesian coordinate of the feed antenna. Transformation from the feed coordinate to the reflectarray coordinates can be accomplished using the rotation functions described in [14]. The three functions rotx(array, α), roty(array, β), and rotx(array, γ) rotate the Cartesian coordinates of the feed horn to the array coordinate system. These transformations are repeated for every element on the reflectarray aperture. Here Nx and Ny are the total number of elements in x and y directions on the reflectarray aperture. Once the incident electric fields on the aperture surface are computed, the reflected fields for every element in the array is then obtained using the generalized scattering matrix, which relates the Cartesian components of incident and reflected fields in the periodic cell, that is, E xref m,n

xx

xy

E xinc m,n

E yref m,n

yx

yy

E yinc m,n

. (4.12)

The scattering matrix is usually obtained from a full‐wave simulation of the reflectarray element with periodic boundary conditions, as discussed in Chapter 2. The phase shift produced for each element depends on the polarization of the feed and, also, takes into account the cross‐polarized reflected fields. In the aperture (reflected) field analysis, the Cartesian components of the tangential electric and magnetic currents on the reflectarray can be obtained using the equivalence principle, as expressed in (4.13):

J sx x ,y

H yref x ,y , J s y x ,y

H xref x ,y , (4.13)

Radiation Analysis Techniques



E yref x ,y , Ms y x ,y

Msx x ,y

E xref x ,y , (4.14)

which give a complete description of the radiating currents of the aperture. Here the magnetic (H) fields are obtained using Maxwell’s equations under local plane wave assumption [15], [16]. From the tangential currents on the reflectarray surface, the far‐ field radiation pattern can be calculated using the vector potentials, which will be described in the next section. 4.2.3  Near‐Field to Far‐Field Transforms and Reflectarray Radiation Pattern The far‐field electric and magnetic fields can be computed using the tangential currents on the aperture as follows: Er

0 jke L 4 r jke jkr L 4 r

E E

0

Hr jkr

N

H

N

H

jke jkr N 4 r jke jkr N 4 r

L

(4.15)

L

Here N and L are functions that are evaluated directly from the tangential currents on the reflectarray surface and are given by

N

J x cos cos

J y cos sin

e jk

xu yv

ds (4.16)

S

N

J x sin

J y cos

e jk

xu yv

ds (4.17)

S

L

M x cos cos

M y cos sin

e jk

xu yv

ds (4.18)

S

L

M x sin

M y cos

e jk

xu yv

ds (4.19)

S

where u and v are the angular coordinates, (u = sinθ cosφ, v = sinθ sinφ). In general, the double integration must be extended to the entire plane z = 0, but it is always limited to the reflectarray surface, because it is assumed that the tangential currents are zero outside the reflectarray. In order to evaluate these integrals element by element, some change of parameters is required. The coordinates of the central point of each reflectarray element can be given by

x

xe

md x

Nx 1 d x ; m 0, 1, 2, 2 Ny 1

, N x 1 (4.20)

d y ; n 0, 1, 2, , N y 1 (4.21) 2 where Nx and Ny are the maximum number of elements in each direction, and dx and dy  represent the periodicity of the reflectarray elements in the x and y directions, y

ye

nd y

85

86

Reflectarray Antennas

respectively. It should be noted that it is assumed that the tangential currents are constant on each element of the reflectarray and also x′e and y′e are limited to the unit‐cell, dy dx dx d y that is, xe , ye . 2 2 2 2 k j N x 1 ud x N y 1 vd y To simplify the formulation, we define the constant K 1 e 2 . Also, as mentioned earlier, since the tangential currents on each reflectarray element are assumed to be constant, they are pulled out of the integrals. The common terms which need to be evaluated for all the integrals can then be expressed in the form of sinc functions. d x /2 d y /2



e

jk xe u ye v

dse

d x d y sin c

d x /2 d y /2

kvd y kud x sin c . (4.22) 2 2

The functions N and L can then be simplified to N

,

K 1 d x d y sin c .

N

,

L

,

L



,

J x xe ,ye cos cos

Nx Ny m 1n 1

Nx Ny m 1n 1

.

Nx Ny m 1n 1

J y xe ,ye cos

e

e

jk mud x nvd y

M y xe ,ye cos sin

e

jk mud x nvd y

kvd y kud x sin c 2 2

M x xe ,ye sin

M y xe ,ye cos

e

(4.23)

(4.24)

kvd y kud x sin c 2 2

M x xe ,ye cos cos

K 1 d x d y sin c

J y xe ,ye cos sin

jk mud x nvd y

kvd y kud x sin c 2 2

J x xe ,ye sin

K 1 d x d y sin c .



m 1n 1

K 1 d x d y sin c .



Nx Ny

kvd y kud x sin c 2 2

jk mud x nvd y

(4.25)

(4.26) .

It should be noted that this summation is extended to a rectangular grid with Nx and Ny elements, but for circular or elliptical reflectarrays, the tangential fields for the elements outside the aperture boundary have to be set to zero. The formulation presented here is the general aperture field analysis method, where the first principal of equivalence is used to obtain the currents on the reflectarray aperture and the corresponding far‐field radiation pattern. It is worthwhile to mention that to simplify the calculations usually only the electric field components are used in the analysis, which corresponds to the second principle of equivalence. Alternatively, the magnetic field components may also be used to compute the far‐field radiation pattern of the reflectarray antenna. More discussion on this topic will be given in a later section.

Radiation Analysis Techniques

In summary, the advantages and disadvantages of the aperture field approach are: Advantages: Accurate modeling of feed and element polarization. Limitations: ●● Complicated formulation and program development. ●● Increased computational time. ●●

4.3 ­Important Topics in Reflectarray Radiation Analysis 4.3.1  Principal Radiation Planes The previous sections discuss calculating the radiation pattern of the antenna in the entire space. However, it is interesting to view the radiation pattern in the principal planes with respect to the main beam direction [17]. While for a broadside radiation the principal planes are the xz‐ and yz‐planes, for off‐broadside beams the principal plane positions do not lie on the spherical coordinate axis. Consider the coordinate system in Figure 4.3, which shows a main beam in an arbitrary direction of (θm, φm). It is necessary to choose the plane cuts that best capture the features of the 3D patterns. For any arbitrary beam direction (θm, φm) it is possible to align the axis of the ­coordinate system with the main beam direction by defining two coordinate rotations. First a rotation of γ = φm about the z‐axis, and next a rotation of β = θm about the y‐axis. By using these two rotations the main beam will be in the direction of z″‐axis in the new  coordinate system (x″, y″, z″). These coordinate rotations are shown pictorially in Figure 4.4. In this new coordinate system, the principal planes (P.P.1, P.P.2) are simply defined as x″z″ and y″z″ planes, respectively. The 2D radiation patterns can then be plotted using the principal plane angles defined as α1 and α2 for P.P.1 and P.P.2. The principal planes and the corresponding angles are shown in Figure 4.5. The task here is to calculate the principal plane angles in terms of the initial coordinate system. As discussed earlier two coordinate rotations are used here. These transformations can be expressed in terms of matrix multiplications which transform the original coordinate system (x, y, z) to the new coordinate system (x″, y″, z″). Figure 4.3  Far‐field coordinate system for the reflectarray antenna.

Z

θm

Y φm

X

87

88

Reflectarray Antennas Z″

Z' Z′ β = θm θm Y″ Y′

X′

Y X'

γ = φm X

X″

(b)

(a)

Figure 4.4  Coordinate rotations used for the reflectarray system: (a) rotation of γ = φm about the z‐axis with Z = Z′, (b) additional rotation of β = θm about the y´‐axis with Y′ = Y″. P.P.1

Z″ P.P.2

α2 α1

Y″

X″

Figure 4.5  Principal planes of the reflectarray antenna.



x y z

cos 0 sin

m

m

0 sin m 1 0 0 cos m

cos m sin sin m cos 0 0

m m

0 0 1

x y . (4.27) z

With this matrix transform a direct relation between the original and new coordinate systems can be obtained as



x y z

cos m x cos m y sin x sin m y coss m sin

m

x cos

m

y sin

m

z sin

m

. (4.28) m

z cos

m

Since the radiation patterns are usually expressed in terms of the spherical or angular coordinates, we can use the formula

Radiation Analysis Techniques

x u sin cos y v sin sin (4.29) z w cos

to obtain the principal plane angles α1 and α2. From Figure 4.5 it can be seen that for P.P.1



x u y 0 z w

sin

cos

sin

1

(4.30) cos

cos

1.

Similarly, for P.P.2



x 0 y v z w

sin cos

sin cos

sin

2

(4.31)

2.

Using (4.30) and (4.31) in (4.28) the spherical or angular coordinates corresponding to the principal plane angles can be calculated. For P.P.1 one obtains



u cos m cos v u tan m .

m

cos

1 tan m

sin

1

tan

m

(4.32)

Similarly, for P.P.2 one obtains u sin

m cos m

sin v u tan m  cos

cos 2 m

2

sin

2

.

sin

m

(4.33)

Equations (4.32) and (4.33) provide a complete description of the principal planes in terms of the angular coordinates (u,v). For any arbitrary beam direction P.P.1 is a straight line passing through the center, while P.P.2 is generally a curved line in the (u,v) plane. For example, the principal planes of a reflectarray antenna with a main beam in the direction of (θm = 26°, φm = 40°) are given in Figure 4.6. 4.3.2  Co‐ and Cross‐Polarized Patterns In the previous sections (4.1 and 4.2), the radiation patterns of the reflectarray were obtained and expressed in terms of θ and φ components. However, in practice it is more common to use the co‐polar and cross‐polar components of the fields according to Ludwig’s third definition [18]. For an x‐polarized feed, the transformation from θ and φ to co‐polar (Eco‐pol) and cross‐polar (Ex‐pol) components are given by Ecox

pol

E xx pol

cos sin

sin cos

E E

, (4.34)

89

Reflectarray Antennas

Figure 4.6  Principal planes of a reflectarray antenna in the angular coordinates with a main beam at the direction of (θm = 26°, φm = 40°).

1

0.5

P.P.2 v

90

0

P.P.1

–0.5

–1 –1

–0.5

0 u

0.5

1

and for a y‐polarized feed by Ecoy

E xy

sin cos

pol pol

E E

cos sin

. (4.35)

For circular polarized feeds, the co‐ and cross‐polar components are defined in terms of the sense of rotation where

1

E RHCP

2

E

jE , E LHCP

1 2

E

jE . (4.36)

The equations provided in this section allow for an accurate calculation of the co‐ and cross‐polar radiation patterns of the reflectarray antenna. This includes all possible sources of cross‐polar radiation from the feed horn, the geometrical projections, and even the reflectarray phasing elements. For the latter case, the cross polarization of the phasing elements has to be obtained from the unit‐cell simulations. 4.3.3  Antenna Directivity Once the radiation pattern of the antenna is obtained from any of the methods described in the previous section, the antenna directivity can be obtained using D0

4 E 2

E

,

m, m 2

2

. (4.37)

sin d d

0 0

Here, (θm,φm) is the direction of the main beam. The calculated directivity in (4.37) takes into account the illumination (taper efficiency) of the array, and the effect of

Radiation Analysis Techniques

projected aperture for off‐broadside beams. Typically, the main challenge in directivity calculations is the evaluation of the denominator in (4.37), which is done numerically. It is important to point out here, that if the radiation patterns are computed in the angular coordinates (u, v) rather than the spherical coordinates (θ,φ), it will be necessary to use Jacobi transformation for the directivity calculations. The differential relation between angular and spherical coordinates is



u sin cos , v sin sin u u v d d , dv d du

v

d .

(4.38)

The matrix relation is then obtained as cos cos sin sin

d d

sin cos cos sin

du . (4.39) dv

The beam solid angle can then be expressed in terms of the angular coordinates as

d

1 cos

sin d d

cos du sin dv

sin du cos dv (4.40)

From this equation, it can be observed that for the case of θ = 90° the beam solid angle shows a singular behavior. Two methods can be used to mitigate this problem where both approaches basically omit the singular points. The first method avoids the calculation of the integral at θ = 90°. The second method removes this singularity in terms of the angular components. The beam solid angle expression in (4.40) is expressed completely in terms of the angular components, that is d



1 cos 1

cos 2 dudv 1 u2

v2

u2 u2

1 sin 2 dv 2 du2 2 v2 uv dudv dv 2 2 2 v u v2

du2

(4.41)

In this method, the singularity is removed for the case where u2 + v2 = 1. It is worthwhile mentioning that since most reflectarrays are designed to generate a beam that is not far off from the broadside direction, the numerical singularity problem is in a region where radiated power is almost zero, hence has trivial effect on the value of the computed directivity. 4.3.4  Antenna Efficiency and Gain Once the reflectarray directivity is calculated, the antenna gain can be obtained by taking into account the aperture efficiency. For an accurate calculation of the reflectarray antenna gain, it is imperative to determine the overall aperture efficiency of the antenna. Different kinds of efficiency factors are considered in reflectarray antennas, and in

91

92

Reflectarray Antennas

general these factors cause a deviation from ideal performance. Some of the most common factors are listed here: ●●

●●

●●

●●

●●

●●

●●

Illumination efficiency is a measure of the uniformity of the feed illumination, where we note that the maximum directivity of an aperture is obtained when the aperture is illuminated uniformly. Spillover efficiency is the percentage of radiated power from the feed that is intercepted by the reflecting aperture, as described in Chapter 3. Phase efficiency is a measure of the uniformity of the phase distribution on the reflectarray aperture. These include the pixelated (unit‐cell) phase distribution on the reflectarray aperture as opposed to a smooth reflector aperture, quantization errors, and limited phase range of the reflectarray elements, and phase wraps on the reflectarray aperture. Material loss such as substrate and conductor loss in microstrip reflectarrays result in an overall reduction of the antenna gain and ultimately the reflectarray efficiency. Polarization efficiency is a measure of the power that is coupled into the cross‐polarized radiated beam of the reflectarray antenna. Blockage efficiency is a measure of the blockage of aperture radiation by the feed assembly, which results in reduction of gain and efficiency in reflectarray antennas. As discussed earlier in Chapter  3, this loss can be completely avoided by using an offset system. Feed efficiency is a measure of the losses in the reflectarray feed system. For a horn antenna feed, these are typically Ohmic losses.

It is important to note that while all these efficiency factors are responsible for the total gain of the reflectarray antenna, as discussed earlier in Chapter 3, the major terms that influence the reflectarray antenna gain are the spillover and taper efficiency. The formulation presented in the previous section for calculating the radiation pattern accurately takes into account several of these factors. Namely illumination efficiency, phase efficiency, and polarization efficiency are already taken into account when one calculates the radiation pattern directivity. Blockage efficiency can also be included in the radiation pattern directivity [9]. The remainder of the efficiency factors, primarily dominated by spillover efficiency and material loss, influence the reflectarray antenna gain. Mathematically this is given as

G

g D. (4.42)

Here D is the maximum directivity of the reflectarray antenna, which is defined as the maximum radiation intensity of the antenna over the average, and is calculated using (4.37), and ηg are the combined efficiency factors that are not included in the computed directivity. The reflectarray antenna gain can also be computed from the far field radiation patterns while using the input power of the feed horn antenna as the reference. The antenna gain is defined as G

,

4

intensity of the antenna in direction power incident at the antenna

,

4

U

,

Pfeed horn

(4.43)

Radiation Analysis Techniques

The radiation intensity is defined as

1 U (θ ,ϕ ) = Re{ E (θ ,ϕ ) × H * (θ ,ϕ )}.r 2 rˆ. (4.44) 2 The gain of the reflectarray antenna can then be evaluated as G



,

E

4

2 2

,

2

0

r

Pfeed horn

, (4.45)

where η0 is the intrinsic impedance of the free‐space. Assuming the losses in the feed horn antenna are negligible, the total power radiated by the feed horn antenna can be computed as EF

2

Pfeed horn

0



2

0

2

,

r 2 sin d d . (4.46)

0

In this equation, EF is the radiation pattern of the horn antenna. For an x‐polarized feed horn, the amplitude squared for the horn radiation pattern is E Fx

,

2

E Fx

E Fx

,

,

*

A0

2

C 2E

cos2

C 2H

sin 2

. (4.47)

Similarly, for a y‐polarized feed horn, the amplitude squared for the horn radiation pattern is E Fy

,

2

E Fy

E Fy

,

,

*

A0

2

C 2E

sin 2

C 2H

cos2

. (4.48)

Assuming that qE = qH = q in the cosine‐q models used for the functions CE and CH, both can be expressed as

E Fx

2

,

E Fy

,

2

2

A0 cos2 q . (4.49)

Substituting this in (4.46) and evaluating the integral, the total power radiated by a linearly polarized feed horn can be calculated as PLP feed horn

A0

2

r2

0



2 . (4.50) 2q 1

Using (4.50) in (4.45), the gain of a linearly polarized reflectarray antenna is given as GLPRA

,

1 A0

2

2q 1 E

,

2

. (4.51)

It is implicit that if the complex constant A0 was taken into account in the calculation of the radiation patterns of the antenna, it will be cancelled out in the gain computations.

93

94

Reflectarray Antennas

For a circularly polarized feed horn, the amplitude of the electric field power is given by

E Fc

2

,

E Fc

E Fc

,

1 A0 2

*

,

2

C 2E

C 2H

, (4.52)

for both right‐ and left‐hand polarizations. Considering that for an ideal circular polarized feed horn qE = qH = q, the total power radiated from the horn antenna can then be calculated as PCP feed horn

2

A0

r2

0



2 . (4.53) 2q 1

Using (4.53) in (4.45), the gain of a circularly polarized reflectarray antenna is given as GCPRA

,

1 A0

2

2q 1 E

2

,

, (4.54)

which as expected is equal to the gain of the linearly polarized antenna (4.51). It is important to note that different methods for computation of reflectarray antenna gain are available in the literature and the interested reader is referred to [19] for a comparative study of these techniques. As a concluding remark, it should be noted that both formulas used for the gain calculations, that is (4.42) and (4.43), correctly take into account the losses due to aperture projection, illumination taper, and spillover. However, (4.43) is more general since in addition to the polarization, the effect of element losses on the antenna gain is also evaluated directly, given that they are included in the radiation pattern calculations. 4.3.5  Spectral Transforms and Computational Speedup In the previous sections it was shown that both radiation analysis approaches, namely array theory and aperture field analysis, require evaluation of a double summation for far‐field calculations [20]. This double summation can be replaced by a 2D inverse discrete Fourier transform (IDFT), defined as f p ,q

1 Nx Ny

N x 1N y 1 m 0 n 0

F m,n e

j

2 mp Nx

e

j

2 nq Ny

. (4.55)

Here the spectral functions will be obtained in a discrete number of angular coordinates. These points in the (u,v) plane are defined by the Fourier transform as: u

v

2 p ; p 0,1,2, N x 1, N x d x k0 (4.56) 2 q ; q 0,1,2, N y 1. N y d y k0

Radiation Analysis Techniques

The main advantage of using spectral functions in the calculations is a significant reduction of computational time, which is simply achieved by replacing the double summations with Fourier transforms. It should be noted that the definition of the propagating wave direction determines whether the discrete Fourier transform or IDFT is to be used for the transforms. In the formulation presented here, we followed the definition in [1], where the IDFT is used to replace the double summation in the spectral functions. It should be noted here that the condition which needs to be satisfied here is that the sets (m,n) and (p,q) must have a one on one relation. This condition can be satisfied by setting the number of far‐field points that are calculated equal to the number of elements on the array. With these constraints defined, it is possible to replace the double summation with the 2D IDFT. By using the 2D inverse fast Fourier transform (2D‐IFFT) algorithm available in Matlab [21], it is possible to calculate the radiation ­patterns of the reflectarrays very efficiently. Some important considerations about implementing the 2D‐IFFT are listed next. 1) The 2D‐IFFT routine described here cannot be implemented if the radiation pattern is being calculated in terms of the spherical coordinates (θ,φ). This is due to the fact that (θ,φ) don’t satisfy the one on one condition with (m,n). However, it is trivial that once the pattern is calculated in the angular coordinates, it is possible to transform it to any other coordinate system. 2) The spectral functions are obtained in a discrete number of angular coordinates. It is possible to make this range symmetric by defining a shift in the variables. The range of (u,v) is typically defined as 0



2d x

u

0

2d x

,

0

2d y

v

0

2d y

, (4.57)

It should be pointed out that if the periodic cell is λo/2, the radiation pattern will be computed in the whole visible range which is defined by the circle u2 + v2 ≤ 1. In addition, note that in this case the absolute value of the range of u and v will be smaller N 1 Ny 1 than 1, due to the x , factors in (4.56). For periodic cells smaller than Nx Ny λo/2, the radiation patterns will be computed in a larger range, while for periodic cells larger than λo/2, the radiation patterns will be computed in a smaller range [1]. 3) The number of points in the (u,v) plane for radiation pattern calculations are equal to the number of elements on the array. For a higher resolution in pattern calculations, it is possible to increase the number of points by extending the grid on the reflectarray surface. This extended grid can be viewed as a virtual reflectarray surface where the amplitude of all the elements outside of the reflectarray is set to zero. With this definition of extended grid, it is possible to obtain high resolution radiation patterns for small size arrays. 4) The implementation of the 2D‐IFFT routine in Matlab also requires some discussion. The FFT routine generally has only one zero frequency point. For the pattern calculations here this point is actually representing the center point in the (u,v) plane, which is (0,0). To satisfy this condition it is necessary to use an odd number of points for radiation pattern calculations. In addition, Matlab’s 2D‐IFFT command swaps the coordinates so that the zero‐frequency point is located at the lower left corner.

95

96

Reflectarray Antennas

After the 2D‐IFFT command (ifft2) is executed, it is necessary to correct the obtained results by using the fftshift command, which corrects the quadrants so that the center of the (u,v) plane is placed in the center of the matrix.

4.4 ­Full‐Wave Simulation Approaches 4.4.1  Constructed Aperture Currents Under Local‐Periodicity Approximation In this approach, the current distribution on the total reflectarray aperture is constructed from unit‐cell analysis of individual elements under local‐periodicity ­ approximations. A common method for determining the currents is based on the spectral domain method of moments (SDMoM). Three different techniques have been proposed to calculate the radiation pattern of printed reflectarrays using this approach, which are briefly summarized here. In the first technique, the radiation pattern is computed from the currents on the array elements with stationary phase evaluation of the spatial dyadic Green’s function (DGF) assuming an infinite ground plane. Despite being slightly more accurate than the array theory and aperture field approaches, the primary drawback of this technique is that the finite size of the reflectarray aperture is not taken into account, and as such diffraction effects are not properly modeled with this approach. The second technique computes the equivalent currents on a surface enclosing the entire reflectarray. The equivalent currents are determined through the Floquet space harmonics from the SDMoM formulation, where the contribution from each array element to the equivalent current is confined to its unit cell. To account for the finite physical size of the array, unit‐cells with no array elements are typically placed at the edges. While the use of equivalent currents on a volume surrounding the reflectarray tends to be more accurate than the first technique, the approximation of the equivalent currents on the unit‐cells using only the fundamental Floquet space harmonics, as well as the discontinuities in the equivalent currents due to truncations at the unit‐cell borders, may result in some errors in computed radiation patterns. The third technique combines the advantages of both approaches by using the equivalent currents approach with a continuous spectrum formulation. In this method, the equivalent currents are constructed on a surface enclosing the entire reflectarray, where the total field on the back side and edges of the reflectarray are assumed to be zero. The tangential fields at the plane of the array are then expanded in a spectrum of plane waves. Note that with this approach, the contribution from each array element over the entire surface is taken into account. This is due to the fact that the electric and magnetic fields at the reflectarray surface are related through the continuous plane wave spectrum and not through the plane wave relation of the fundamental Floquet harmonic as in the second technique. More discussions on the accuracy of these techniques, along with numerical and measured results will be given in the next section. 4.4.2  Complete Reflectarray Models The analysis techniques presented thus far, analyze the radiation performance of the reflectarray antenna under local periodicity approximation. While this approximation is valid to a good degree, accurate analysis of the radiation pattern requires electromagnetic

Radiation Analysis Techniques

modeling of the full array. The challenge here is that the electrically large size of the reflectarray antenna aperture, combined with thousands of elements with dimensions smaller than a wavelength, demands a considerably high computational time and resources for full‐wave analysis. Despite this drawback, the full‐wave analysis approach is generally considered to be the most accurate technique for reflectarray analysis. Many commercial electromagnetic simulation software can be used for analysis of reflectarray antennas. A challenge in geometry modeling of the reflectarray antenna is the variable dimension/rotation of the large number of elements. An efficient way to overcome this is to input the dimensions and geometrical locations of the elements into the software, by generating a data file containing this information in a format that is recognized by the software. The geometrical location of the reflectarray elements can best be specified by the center position of the unit‐cell. Typically, for printed reflectarrays, only the x‐ and y‐locations of each unit‐cell center needs to be specified. As an example, for variable size patches, the coordinate information can be given in the first and second columns, while the next columns contain the width and length of the variable size patches. A very common geometry file format for 2D structures is DXF. The overall file organization of a DXF file contains a header, classes, tables, blocks, entities, and objects sections, and can be used to generate almost any 2D geometry. Here we will briefly outline the format for generating DXF files for square patches, which are the most common type of reflectarray elements. First, the (x, y, z) coordinates of each vertex have to be computed and imported in the given format. While the computation of the vertex positions is straightforward, the vertex positioning has to follow the format order. The vertex positions have to be specified in a manner that creates a closed loop with a counter clockwise order. Namely, that is the top left, bottom left, bottom right, and top right vertices of the square. Using this approach, a full array containing hundreds of elements can be easily imported into any simulation software. However, note that these are geometry files, and once imported the electrical parameters, such as conductivities, have to be specified for full‐wave simulations. An image of variable size square patches for a reflectarray antenna imported into Ansys HFSS is shown in Figure 4.7. Z

Y

0

40

X

80 (mm)

Figure 4.7  Geometry of reflectarray patch elements modeled in Ansys HFSS.

97

Reflectarray Antennas

Once the element geometries have been modeled in the full‐wave software, modeling the substrate and feed antenna is relatively straightforward. Two different full‐wave analyses have been proposed for reflectarrays that are referred to as transmit mode and receive mode analysis techniques. In the transmit mode analysis, the reflectarray aperture is illuminated by the feed antenna and the far‐field radiation characteristics of the antenna is computed using the full‐wave approach. In the receive mode analysis, the reflectarray is excited with a plane wave impinging on the aperture, and the field distribution at the focal region of the array (where the feed antenna phase center would be placed) is analyzed. Examples of full‐wave reflectarray analysis techniques are presented in the next section.

4.5 ­Numerical Examples 4.5.1  Comparison of the Array Theory and Aperture Field Analysis Techniques We consider Ka‐band reflectarray designs with a circular aperture and a diameter of 17λ at the design frequency. The phasing elements, used in this study, are variable size square patches with a unit‐cell periodicity of λ/2 at the design frequency of 32 GHz and are fabricated on a 20 mil Rogers 5880 substrate. The reflection phase response (S‐ curve) of the phasing elements obtained using the infinite array approach, is generated using Ansoft Designer, and is given in Figure 4.8. Note that as discussed in Chapter 2, the reflection characteristics of the phasing elements are angle dependent; however, for this design, normal incidence can present good approximations for oblique incidence angles up to 30o. Thus, the reflectarrays here are designed based on the simulated reflection coefficients obtained with normal incidence. In addition. the element losses and cross‐polarization are also negligible here, hence are ignored in the computations. 150 Reflection Phase (degrees)

98

θi = 0°

100

θi = 30° (TM)

50

θi = 30° (TE)

0

–50 –100 –150 –200

1

1.5

2

2.5

3

3.5

4

4.5

Patch width (mm)

Figure 4.8  Reflection phase versus patch width for the reflectarray elements.

Radiation Analysis Techniques

4.5.1.1  Example 1: Reflectarray Antenna with a Broadside Beam

We first consider an axisymmetric design. The reflectarray phasing elements are designed to generate a beam in the broadside direction. An x‐polarized prime‐focus feed horn is positioned with an F/D ratio of 0.735. For the horn model used in this study, the power q of the feed radiation pattern is 6.5 at 32 GHz. As discussed previously, in the array theory calculations, the polarization of the feed horn is not modeled. The aperture taper and the ideal phase requirement for the reflectarray elements are given in Figure 4.9. For this system, the spillover and illumination efficiency are 93.01% and 81.98%, respectively. This corresponds to an aperture efficiency of 76.25%. In the next stage, the dimensions of the patch elements are selected from the S‐curve in Figure 4.8 to match the required phase distribution on the aperture. The mask of the reflectarray antenna and the obtained reflection phase of the elements are given in Figure 4.10. It can be seen

–5 20 –10

10

10

(a)

20

30

y-axis [element number]

y-axis [element number]

0 30

30 100 20

0

10

–100

–15

10

(b)

x-axis [element number]

20

30

x-axis [element number]

y-axis [element number]

Figure 4.9  Reflectarray design with a broadside beam: (a) Aperture taper, (b) Phase requirement on the aperture.

30 100 20

0

10

–100

10

(a)

(b)

20

30

x-axis [element number]

Figure 4.10  Reflectarray design with a broadside beam: (a) Mask of the reflectarray antenna, (b) Reflection phase of the elements on the reflectarray aperture.

99

Reflectarray Antennas

Radiation Pattern (dB)

0

Array Theory Aperture Field (co-pol)

–10 –20 –30 –40 –50 –60

–50

0

50

α1 (degrees)

(a) 0 Radiation Pattern (dB)

100

Array Theory Aperture Field (co-pol)

–10 –20 –30 –40 –50 –60

(b)

–50

0

50

α2 (degrees)

Figure 4.11  Radiation pattern of the reflectarray with a broadside beam: (a) xz‐plane, (b) yz‐plane. Source: Nayeri 2013 [20]. Reproduced with permission of IEEE.

the phase distribution obtained from the variable size patch elements (Figure 4.10 (b)) shows a close agreement with the ideal phase, that is Figure 4.9 (b). Now that all the system parameters are determined, the radiation performance of the reflectarray antenna can be computed using the analysis approaches described in the previous section. The principal plane (P.P.1 and P.P.2) radiation patterns of the reflectarray antenna, calculated by both methods are given in Figure 4.11 at 32 GHz. It should be noted that with this design the cross‐polarized pattern obtained using the aperture field formulation is almost zero in the principal planes. The maximum cross‐polarization level for this system is −36.1 dB, which occurs in the 45° planes. A contour plot of the radiation patterns obtained by the aperture field analysis approach is also given in Figure 4.12. 4.5.1.2  Example 2: Reflectarray Antenna with an Off‐Broadside Beam

For the second design, we consider an offset‐fed reflectarray with an off‐broadside beam. The phasing elements are designed to generate a beam in the direction of (θ,  φ) = (25°, 0°). The offset feed horn position is Xfeed = −45.9 mm, Yfeed = 0 mm,

Radiation Analysis Techniques 1

1

–36.5

–5 –10

0.5

–37

0.5

–37.5

–20

v

0

v

–15

–38

0

–38.5

–25 –0.5 –1 –1

–30

–0.5

–35 –0.5

(a)

0 u

0.5

–39 –39.5

–1 –1

1

1

0 u

(b)

Figure 4.12  Radiation pattern of the reflectarray with a broadside beam: (a) co‐pol, (b) cross‐pol.

30 –5 20 –10

10

10

(a)

20

30

x-axis [element number]

y-axis [element number]

y-axis [element number]

0

–15

30 100 20

0

10

–100

10

(b)

20

30

x-axis [element number]

Figure 4.13  Reflectarray design with an off‐broadside beam: (a) Aperture taper, (b) Phase requirement on the aperture.

Zfeed = 98.4 mm. The feed horn is LHCP and the power q of the feed radiation pattern is 6.5 at 32 GHz. The aperture taper and the ideal phase requirement for the reflectarray elements are given in Figure 4.13. For this system, the spillover and illumination efficiency are 92.19 and 77.92%, respectively. This corresponds to an aperture efficiency of 71.83%. Similar to the previous design, the dimensions of the patch elements can now be selected from the S‐curve in Figure 4.8 to match the required phase distribution on the aperture. The mask of the reflectarray antenna and the obtained reflection phase of the elements are given in Figure 4.14. Similarly, it can be seen the phase distribution obtained from the variable size patch elements show a close agreement with the ideal phase, that is, Figure 4.13(b). As discussed earlier, in the aperture field approach the polarization of the feed is modeled in the analysis. As such it would be interesting to observe the field distribution on the aperture for both incident and reflected fields. The incident electric fields on the reflectarray aperture are given in Figure 4.15. To generate a collimated beam, the reflected phase, after element phase compensation, should be flat along the y‐axis and progressive in the direction of the scanned beam (x‐axis). The phases of the

101

Reflectarray Antennas

y-axis [element number]

30 100 20 0

10

–100

10

(a)

(b)

20

30

x-axis [element number]

Figure 4.14  Reflectarray design with an off‐broadside beam: (a) Mask of the reflectarray antenna, (b) Reflection phase of the elements on the reflectarray aperture. 0

–5 20

–10

10

–15

10

(a)

30

100

0

10

–100

20

20

–10

10

–15

10

30

x-axis [element number]

–5

(b)

30

20

30

–20

x-axis [element number]

10

(c)

20

y-axis [element number]

30

y-axis [element number]

y-axis [element number]

0

y-axis [element number]

102

30

–20

x-axis [element number]

30 100 20

0

10

–100

10

(d)

20

20

30

x-axis [element number]

Figure 4.15  Incident electric fields on the reflectarray aperture: (a) |Ex| (dB), (b) |Ey| (dB), (c) phase of Ex, (d) phase of Ey.

30 100 20

0

10

–100

10

(a)

20

y-axis [element number]

y-axis [element number]

Radiation Analysis Techniques

30

x-axis [element number]

30 100 20

0

10

–100

10

(b)

20

30

x-axis [element number]

Figure 4.16  Reflected electric fields on the reflectarray aperture: (a) phase of Ex, (b) phase of Ey.

reflected electric fields are shown in Figure 4.16 where indeed such a phase distribution can be observed. It should be noted that the reflectarray elements used in the design here have a reflection coefficient magnitude close to unity, thus the magnitude of the reflected field remains almost the same as that of the incident field, and are not shown here for brevity. The radiation patterns of this offset reflectarray antenna is also computed using both analysis approaches. The radiation patterns in the principal planes at 32 GHz are given in Figure 4.17, where again note that there is a very good agreement between both analysis techniques. For this offset system, the cross‐polarized radiation pattern is also observed in both principal planes, where the maximum cross‐polarization level is −30.0 dB. 4.5.1.3  Comparison of Calculated Directivity versus Frequency

From the results given in Figure 4.11 and Figure 4.17 for the two reflectarray systems, it can be seen that the calculated radiation pattern obtained by both methods are in close agreement with each other. In particular, the main beam direction, beamwidth, and general pattern shape are almost identical, however, a slight difference is observed in the side‐lobe regions. The antenna directivity is a suitable measure to compare the calculated radiation performance of these methods. In order to accurately model the reflectarray directivity versus frequency, the frequency behavior of the feed horn pattern and the element reflection characteristics are implemented into this calculation routine. For the phasing elements, the frequency behavior of the reflection phase is obtained across the band from full‐wave simulations. For the horn model used in this study, the power q of the feed radiation pattern varies almost linearly from 5 at 30 GHz to 8.3 at 34 GHz according to the measured data. For the two reflectarray systems studied here, the directivity versus frequency is given in Figure 4.18. It can be seen that the computed directivity versus frequency obtained by both methods also shows a close agreement. It should be noted here that, as expected, the off‐broadside system shows a lower directivity. At the center frequency of 32 GHz, the difference in computed directivity is less than 0.1 dB for both designs. It is worth pointing out that a possible reason for the slightly lower directivity obtained with the

103

Reflectarray Antennas

Radiation Pattern (dB)

0 –10

Array Theory Aperture Field (co-pol) Aperture Field (x-pol)

–20 –30 –40 –50 –60

–50

0

50

α1 (degrees)

(a) 0

Array Theory

Radiation Pattern (dB)

Aperture Field (co-pol)

–10

Aperture Field (x-pol)

–20 –30 –40 –50 –60

–50

0

50

α2 (degrees)

(b)

Figure 4.17  Radiation pattern of the reflectarray with an off‐broadside beam: (a) P.P.1, (b) P.P.2. Source: Nayeri 2013 [20]. Reproduced with permission of IEEE. 34 33

Directivity (dB)

104

Array Theory Aperture Field

32

broadside design off-broadside design

31 30 29 28 27 28

30

32

34

36

Frequency (GHz)

Figure 4.18  Directivity versus frequency for the two reflectarray systems. Source: Nayeri 2013 [20]. Reproduced with permission of IEEE.

Radiation Analysis Techniques

array theory approach is the effect of the element receive mode pattern. In the aperture field method, the reflection coefficients of the elements are obtained under normal incidence approximation. In other word, an omni‐directional element receive pattern is assumed in the aperture field analysis. 4.5.2  Consideration in the Array Theory Technique: Element Pattern Effect In the array theory formulation, the radiation pattern of the reflectarray antenna is computed using (4.7), which takes into account the element receive (Rx) and transmit (Tx) patterns. As discussed earlier, the elements are modeled with a cosine qe radiation pattern, where in many cases qe =1. Meanwhile an isotropic element pattern is also used for the elements of large reflectarrays. Therefore, it is interesting to observe the effects of element pattern on the radiation pattern of the reflectarray antenna. To study this effect, we consider the same off‐broadside system studied in the previous section. The radiation pattern in the principal plane is given in Figure 4.19 for four different cases. From these results, it can be seen that although the radiation patterns show a close agreement around the area of the main beam, a noticeable difference in the pattern shape is observed in the off‐main beam areas. In particular, the element transmit pattern plays an important role here, where in comparison almost 10 dB difference is observed in the minor lobes. Similar results were observed in the other principle plane. In addition, note that while the difference is quite small, when the element transmit mode pattern is ignored, the antenna beam width is slightly wider, which would also correspond to a lower computed directivity. This study demonstrates the importance of the element pattern shape in the radiation analysis of reflectarray antennas. While for the square type patch elements studied here the element pattern was a simple cosine model, accurately determining the ­element pattern shape is essential before one computes the radiation pattern of a reflectarray antenna.

Radiation Pattern (dB)

0 –10 –20

With Complete Element Pattern With Isotropic Element Rx Pattern With Isotropic Element Tx Pattern With Isotropic Element Pattern

–30 –40 –50 –60

–50

0

50

α1 (degrees)

Figure 4.19  Radiation pattern of the reflectarray antenna and the effect of element pattern in P.P.1.

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Reflectarray Antennas

4.5.3  Consideration in the Aperture Field Technique: Variations of Equivalence Principle The aperture field approach is based on the principle of equivalence. It was discussed earlier that the radiation pattern of the antenna can be obtained using any of the three variations of equivalence principle, that is, both electric and magnetic currents, only electric currents, and only magnetic currents. To study this numerically, we consider the same off‐broadside reflectarray system. The radiation pattern in the principal plane is given in Figure 4.20 for the three cases. Similar results were observed in the other principle plane. These results show that as expected the co‐polarized radiation pattern obtained by these three methods are almost identical, hence either approach can be applied. On the other hand, some difference is observed in the cross‐polarized radiation patterns. In general, using both electric and magnetic fields on the aperture yields a better representation of the problem, and as such it is expected that using the first equivalence of principle is the most accurate analysis approach. Comparison with full‐wave simulations will be presented in the next section, which further validates this discussion.

Radiation Pattern (dB)

0 –10

Using Both Js and Ms Using Only Js Using Only Ms

–20 –30 –40 –50 –60

–50

0

50

α1 (degrees)

(a) 0 Radiation Pattern (dB)

106

–10

Using Both Js and Ms Using Only Js Using Only Ms

–20 –30 –40 –50 –60

(b)

–50

0

50

α1 (degrees)

Figure 4.20  Radiation pattern of the reflectarray antenna and the effect of aperture currents in equivalence principle: (a) P.P.1, (b) P.P.1 zoomed view.

Radiation Analysis Techniques

4.5.4  Comparisons with Full‐Wave Technique As discussed earlier, full‐wave techniques provide the most accurate solution to the reflectarray radiation problem. In this section, we compare the accuracy of the reflectarray radiation pattern computed using earlier approaches by comparing them with full‐wave simulations. Note that for this comparative study, full‐wave simulations will be more advantageous than measured results, since measurement results are susceptible to both fabrication and measurement errors. Due to the limited computational resources, we consider a smaller Ka‐band reflectarray with a circular aperture and a diameter of 14.5λ at the design frequency of 32 GHz. The feed is positioned at Xfeed = −45.90 mm, Yfeed = 0 mm, Zfeed = 98.44 mm based on the coordinate system. The elements phases are designed to generate a beam in the direction of (θ, φ) = (25°, 0°). Similarly, for the reflectarray phasing elements, the variable size square patches are selected from the S‐curve data in Figure  2.23. The 609‐element reflectarray antenna is modeled using the commercial electromagnetic software FEKO. For the excitation of the reflectarray, a point source feed model with a cos6.5θ radiation pattern is used. The advantage of using a point source rather than a feed horn here is that a point source model does not have a blockage aperture, which makes it more suitable for comparison purpose, since blockage is typically not modeled in the classical methods. For this design 568,435 unknown basis functions have to be calculated for the FEKO method of moments (MoM) solution. Considering the large number of unknowns, the multilevel fast multi‐pole method (MLFMM) solver in FEKO was selected for this simulation. The geometry of the reflectarray antenna modeled in FEKO and the simulated 3D radiation pattern are shown in Figure  4.21. The full‐wave simulation here take into account all approximations in reflectarray element design and mutual

(a)

(b)

Figure 4.21  Reflectarray antenna simulated using FEKO, (a) top view of the reflectarray, (b) 3D radiation pattern of the reflectarray. Source: Nayeri 2013 [20]. Reproduced with permission of IEEE.

107

Reflectarray Antennas 0

Radiation Pattern (dB)

–10

0

Aperture Field (co-pol) FEKO (co-pol) FEKO (x-pol)

–20

–20 –40

20

30

–30 –40 –50 –60

–50

0

50

α1 (degrees)

(a) 0 –10

Radiation Pattern (dB)

108

0

Aperture Field (co-pol) Aperture Field (x-pol) FEKO (co-pol) FEKO (x-pol)

–20

–20 –40 –10

0

10

–30 –40 –50 –60

(b)

–50

0

50

α2 (degrees)

Figure 4.22  Radiation patterns of reflectarray antenna: (a) P.P.1, (b) P.P.2. Source: Nayeri 2013 [20]. Reproduced with permission of IEEE.

coupling, as well as the edge diffraction effects. Therefore, comparing these simulation results with the results obtained using classical approaches can provide a good measure in terms of the accuracy of these approaches. The principal plane (P.P.1 and P.P.2) radiation patterns of the reflectarray antenna calculated using the aperture field method and the full‐wave simulation are given in Figure  4.22. The computed directivity and gain of the reflectarray antenna using the aperture field approach is 30.48 dB and 29.68 dB, respectively. To accurately model the cross‐polarization of the reflectarray antenna in the aperture field analysis, the incident fields on the reflectarray aperture obtained from the full‐wave simulations were used in these calculations. It should be noted that the calculated co‐polarized radiation pattern of the reflectarray antenna obtained using the array theory approach was similar to the aperture field results, and is not shown in this section for the sake of brevity.

Radiation Analysis Techniques

Comparison of the results given here show that the aperture field analysis approach accurately calculate the general pattern shape, main beam direction, beam‐width, and the side‐lobe and cross‐polarization level in the main beam area. Outside the main beam area, however, some discrepancies are observed between these results. These are primarily due to element design approximations and edge diffraction effects that are not taken into account in the analysis approaches. In particular, comparison between the ideal phase shift and the phase shift obtained by the reflectarray elements, Figure 4.23, indicates that while the reflectarray aperture does indeed generate a phase shift to collimate the beam, this difference in phase shift is the primary reason for the discrepancies in the radiation pattern. To better observe the difference between the ideal phase shift and the phase shift obtained by the reflectarray elements, the phase distribution along the x‐axis (y = 0) is also given in Figure 4.24. 350 300

300

250

250

200

0

350

50

150

y (mm)

y (mm)

50

200

0

150 100

100 50

–50 –50

(a)

0

50

50

–50

0

–50

0 x (mm)

(b)

x (mm)

50

Figure 4.23  Phase shift on the reflectarray aperture, (a) ideal phase shift, (b) phase shift obtained by the reflectarray elements. Source: Nayeri 2013 [20]. Reproduced with permission of IEEE. 450 Ideal Patch Elements

400

Phase Shift (deg.)

350 300 250 200 150 100 50 0 –100

–50

0

50

100

x-axis (mm)

Figure 4.24  The progressive phase shift on the reflectarray aperture.

109

110

Reflectarray Antennas

In conclusion, while it is implicit that a full‐wave simulation will provide accurate results, the main disadvantage of full‐wave simulations is the high computational time and resources required. In total, the full‐wave simulation here required 29.56 GB of memory with a CPU time of 26.97 hours on an 8 core 2.66 GHz Intel(R) Xeon(R) E5430 computer. In comparison, the CPU time for the array theory and aperture field calculations were 1.78 s and 46.4 s on the same computer using a single core, respectively. It is worth pointing out that, in general, the classical approaches have limited accuracy and when accurate radiation pattern computation is required, full‐wave simulation of the reflectarray system is necessary.

­References 1 J. Huang and J. A. Encinar, Reflectarray Antennas. New York, NY, USA: Wiley‐IEEE, 2008. J. Huang and R. J. Pogorzelski, “A Ka‐band microstrip reflectarray with elements having 2

3 4

5

6

7

8

9 10 11 12 13 14

variable rotation angles,” IEEE Trans. Antennas Propag., Vol. 46, No. 5, pp. 650–656, May. 1998. D. M. Pozar, S. D. Targonski, and H. D. Syrigos, “Design of millimeter wave microstrip reflectarrays,” IEEE Trans. Antennas Propag., Vol. 45, No. 2, pp. 287–296, Feb. 1997. Y. Zhuang, K. Wu, C. Wu, and J. Litva, “A combined full‐wave CG‐FFT method for rigorous analysis of large microstrip antenna arrays,” IEEE Trans. Antennas Propag., Vol. 44, No. 1, pp. 102–109, Jan. 1996. Y. Li, M. E. Bialkowski, K. H. Sayidmarie, and N.V. Shuley, “81‐element single‐layer reflectarray with double‐ring phasing elements for wideband applications,” IEEE Antennas and Propagation Society International Symposium, Toronto, Canada, July 2010. H. Rajagopalan, S. Xu, and Y. Rahmat‐Samii, “Graphical visualization of reflectarray reflection phase response,” IEEE Antennas and Propagation Society International Symposium, Washington, USA, July 2011. J. A. Encinar and J. A. Zornoza, “Three‐layer printed reflectarrays for contoured beam space applications,” IEEE Trans. Antennas Propag., Vol. 52, No. 5, pp. 1138–1148, May 2004. P. Nayeri, F. Yang, and A. Z. Elsherbeni, “Single‐feed multi‐beam reflectarray antennas,” IEEE Antennas and Propagation Society International Symposium, Toronto, Canada, July 2010. Y. Rahmat‐Samii, “Reflector Antennas,” in Antenna Handbook: Theory, Applications, and Design, Y. T. Lo and S. W. Lee, Van Nostrand Reinhold, 1988. C. J. Sletten, Reflector and Lens Antennas: Analysis and Design Using Personal Computers, Artech House Inc., 1988. M. E. Cooley, and D. Davis, “Reflector antennas,” in Radar Handbook, 3rd Edn, McGraw‐ Hill, 2008. A. D. Olver, P. J. B. Clarricoats, A. A. Kishk, and L. Shafai, Microwave Horns and Feeds, The Institution of Electrical Engineers, 1994. Y. Rahmat‐Samii, “Useful coordinate transformations for antenna applications,” IEEE Trans. Antennas Propag., pp. 571–574, July 1979. L. Diaz and T. Milligan, Antenna Engineering Using Physical Optics, Norwood, MA: Artech House, 1996.

Radiation Analysis Techniques

15 W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, 3rd Edn, Hoboken, NJ:

John Wiley & Sons Inc., 2012.

16 C. A. Balanis, Antenna Theory: Analysis and Design, 3rd Edn, Hoboken, NJ: John Wiley &

Sons Inc., 2005.

17 A. Yu, “Microstrip reflectarray antennas: Modeling, design and measurement,”

Ph.D. dissertation, Dept. Elect. Eng., University of Mississippi, Oxford, MS, 2010.

18 A. Ludwig, “The definition of cross polarization,” IEEE Trans. Antennas Propag., Vol. 21,

No. 1, pp. 116–119, Jan. 1973.

19 B. Devireddy, A. Yu, F. Yang, and A. Z. Elsherbeni, “Gain and bandwidth limitations

of reflectarrays,” ACES Journal, Vol. 26, No. 2, pp. 170–178, Feb. 2011.

20 P. Nayeri, A. Z. Elsherbeni, and F. Yang, “Radiation analysis approaches for reflectarray

antennas,” IEEE Antennas Propag. Magazine, Vol. 55, No. 1, pp. 127–134, Feb. 2013.

21 Matlab, The Mathworks Inc., Natick, MA.

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113

5 Bandwidth of Reflectarray Antennas As discussed in the earlier chapters, the reflectarray combines the numerous advantages of both antenna arrays and reflectors and creates a low‐profile, low‐mass, and low‐cost antenna with a high gain. Despite the numerous advantages this hybrid configuration can offer, the reflectarray antenna suffers from a major limitation: an inherently narrow bandwidth. The bandwidth of reflectarray antennas depends on its element, aperture size, focal length, and so on, and typically does not exceed beyond 10%. In this chapter, we will study the fundamental physics that determine the bandwidth of reflectarray antennas.

5.1 ­Bandwidth Constraints in Reflectarray Antennas 5.1.1  Frequency Behavior of Element Phase Error While reflectarrays mimic the traditional reflector antennas [1], there is an inherent fundamental difference between the two when it comes to system bandwidth [2]–[5]. A reflector antenna is essentially frequency‐independent, that is, it has an infinite bandwidth, and in practice the primary feed antenna is responsible for limiting the bandwidth of the reflector. On the other hand, a reflectarray suffers from bandwidth limitations due to the physical nature of its phasing elements and also the flat nature of its aperture. Both factors introduce phase errors as a function of frequency, which limit the bandwidth of reflectarray antennas. For microstrip reflectarrays, the element itself has a bandwidth of typically only a few percent, ultimately resulting in a narrow bandwidth for the reflectarray system. Techniques such as using thick substrates, stacked patches, or multi‐resonant elements can improve the bandwidth of the elements and the reflectarray antenna, and bandwidths in excess of 20% have been reported. Regardless of the choice of phasing elements used for the design, reflectarray elements are usually selected to satisfy the required phase at the design frequency. As the frequency changes, the required phase in the array will change, and the reflectarray element phase will also change. If these two phase changes do not match with each other, which is usually the case, it will introduce phase errors in the array, resulting in pattern deterioration, gain reduction, and ultimately, bandwidth limitations of the antenna. Reflectarray Antennas: Theory, Designs, and Applications, First Edition. Payam Nayeri, Fan Yang, and Atef Z. Elsherbeni. © 2018 John Wiley & Sons Ltd. Published 2018 by John Wiley & Sons Ltd.

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Reflectarray Antennas

It’s worthwhile pointing out that the phase difference between reflectarray elements is important here rather than the reflection phase of an individual element, since in a reflectarray design, the phase difference between the elements is important for obtaining the desired radiation pattern. Consequently, defining an element bandwidth for reflectarray phasing elements is not practical without considering the relative phase requirements of all the reflectarray elements. It is, however, possible to study the phase error by considering the frequency behavior of the reflectarray elements for certain relative phase requirements. Without loss of generality, we first consider a reflectarray with a main beam at the broadside direction (ri . rˆo = 0). The geometrical center of the reflectarray surface (point 1) is selected as a phase reference, and the center point of an arbitrarily element (point 2) is selected on the reflectarray surface to explain the phase relation. The phase difference between these two points at the center frequency (  f0) satisfies the following relation

2

f0

f0

1

k0 R2 R1 . (5.1)

As the frequency changes, the phase difference will also change. An ideal phase relation is

2

f

1

f

k R2 R1 . (5.2)

Combining (5.1) with (5.2), one can obtain 2



f

1

f

2

f0

f

1

f0

f0

2 c

R2 R1 . (5.3)

It is clear from these equations that the phase difference ( ( f ) 2( f ) 1 ( f )) should be a function of frequency. When the frequency increases, the phase difference should also increase. In practice, this ideal frequency behavior of the phase difference cannot be satisfied, and phase errors will occur with frequency, which reduces the reflectarray gain and narrows the reflectarray bandwidth. To study this effect quantitatively, a phase error term is defined as follows:

PhaseError f

2

f

1

f

k R2 R1 . (5.4)

In this equation, the first term represents the frequency behavior of the elements, while the second term is the required spatial phase delay. In total, this equation represents the phase error of elements relative to a certain phase requirement. Thus, it provides a good measure to compare different element designs: the smaller the phase error, the wider the reflectarray bandwidth. It should be pointed out that in this phase error formula, the quantized phase of the reflectarray elements will result in a non‐zero phase error at the center frequency (  f0), however, quantization phase error is usually small in comparison. To quantitatively observe the phase errors introduced on the aperture of a reflectarray antenna, we consider a case where two elements on the aperture are required to have a 90° relative phase difference at the center design frequency of 32 GHz:

200

1

100

0.995

0

0.99

–100 –200 (a)

|Γ|

∠Γ (deg.)

Bandwidth of Reflectarray Antennas

0.985

1

2

(b)

3 4 Patch Width (mm)

5

0.98

Figure 5.1  A Ka‐band reflectarray variable‐size square patch element: (a) geometry, (b) reflection coefficients at 32 GHz.



2

f0

1

f0

k0

R2

R1

90 . (5.5)

The element chosen for this study is a variable‐size square patch, which was studied in detail in Chapter 2. The patch is fabricated on a 20 mil Rogers Duroid 5880 laminate where the unit‐cell size is 4.7 × 4.7mm2. The reflection coefficients as a function of patch width are given in Figure  5.1. Using this element data, for a 0° reflection phase, the patch width should be 2.69 mm; for a 90° reflection phase, it corresponds to a patch size of 2.41 mm. Figure 5.2(a) shows how the reflection phases of these 0° and 90° elements vary with frequency. As discussed previously, it is the frequency behavior of the phase difference that determines the reflectarray bandwidth. Thus, the phase error curve, as defined in (5.4), is also given in Figure 5.2(b). At the center frequency 32 GHz, the phase error is zero. As the frequency deviates from the center frequency, the phase error increases. At 30 GHz and 34 GHz, the phase error is −41.02 and 35.21°, respectively. 5.1.2  Frequency Behavior of Spatial Phase Delay The bandwidth limitation imposed by the radiating element is most significant for small or moderate size reflectarrays, and bandwidth of large reflectarrays is primarily governed by a second factor, namely the differential spatial delay. For the space‐fed system of the reflectarray antenna, the different path lengths from the feed phase center to each element on the aperture result in differential spatial delays for the elements. In order to illustrate this, we look at the cross‐sectional view of a generic front‐fed axisymmetric reflectarray antenna as shown in Figure 5.3. The rays emanating from the phase center of the feed antenna expand spherically. We show two rays traveling the paths R1 and R2, where the former travels the distance from the feed phase center to the reflectarray aperture center, and the latter travels from the same starting point to the reflectarray edge. ΔR is the path difference between the two rays. Depending on the system setup, ΔR can be several wavelengths at the center design frequency. Mathematically this can be written as

R

N n

, (5.6)

115

Reflectarray Antennas

150

Ideal phase change for a relative 90° phase difference at 32 GHz

∠Γ (degrees)

100 50

90 degrees

0 0 degrees –50 –100 30

Patch width = 2.69 mm Patch width = 2.41 mm 31

(a)

32

33

34

Frequency (GHz) 40

Phase Error (degrees)

116

20 0 –20 –40 –60 30

(b)

31

32

33

34

Frequency (GHz)

Figure 5.2  (a) Reflection phase as a function of frequency for two patch widths. (b) Phase error as a function of frequency. Feed phase center

R1

R2

∆R

Reflectarray

Figure 5.3  Cross sectional view of a reflectarray system and path delays.

Bandwidth of Reflectarray Antennas 350 300

80

250

60

200 150

40

100 20

50 20 40 60 80 100 x-axis [element number]

(a)

2500

100 y-axis [element number]

y-axis [element number]

100

0

(b)

2000

80 60

1500

40

1000

20

500

20 40 60 80 100 x-axis [element number]

0

Figure 5.4  Phase distribution on a reflectarray antenna aperture in degrees: (a) truncated to one phase cycle, (b) no truncation.

where λ is the wavelength at the center frequency, N is an integer, and n is a fractional number. In a reflectarray antenna, typically the phasing element is designed to compensate for the nλ term, by providing a phase shift between 0 and 2π. To further illustrate this, we consider an axisymmetric reflectarray with a circular aperture and a diameter of 50 wavelengths. The feed is placed at a distance of 40 wavelengths from the center of the aperture and is pointing to the geometrical center. The phase distributions on the aperture are given for two cases in Figure 5.4. In the first case, the phases are truncated to provide a phase shift only between 0 and 2π. In the second case, the true physical difference is considered to determine the phase without wrapping, where it can be seen that this corresponds to several phase cycles for the majority of elements. To understand the frequency behavior of the spatial phase delay, we consider an off‐center frequency with the wavenumber given as

k1

2 0

. (5.7)

When the 2π phase truncation was implemented, as shown in Figure 5.4(a), the phase compensation was designed for each element at the center design frequency. This, however, results in a phase error term at off‐center frequencies that is mathematically given by

Phase Error

k1 k0

R. (5.8)

Depending on the design, this phase error could be very large portions of a full 2π phase cycle, and is the primary bandwidth limitation factor for large reflectarray antennas or moderate size reflectarrays with small f/D ratios. For the 50 wavelength reflectarray studied here the phase wraps on the aperture and the phase error at an off‐center frequency of Δλ = −0.1λ is given in Figure 5.5(a) and (b), respectively. Note that for this design, six complete phase wraps are observed on the reflectarray aperture. It can be seen that as a result of frequency change a phase error is generated on the aperture which moves outward from the center of the reflectarray aperture. In other

117

Reflectarray Antennas

Phase Error (deg.)

60 40 20

20

(a)

6

250

80

40

60

80

x-axis [element number]

5

200

4

150

3

100

2

50

1

0

100

20

(b)

40

60

80

100

Number of Phase Wraps

100 y-axis [element number]

118

0

x-axis [element number]

Figure 5.5  (a) Phase wraps on the aperture of the reflectarray. (b) Phase errors on the aperture of the reflectarray along the center x‐axis at an off‐center frequency with Δλ = −0.1λ0. A 180° out of phase is observed when N = 4.

words, the maximum phase error will be observed at the edges of the aperture. The phase error plot clearly shows that as frequency deviates further from the center design frequency, the phase errors on the aperture increase significantly. Further, note that toward the edges of the reflectarray aperture the phase error approaches 180°, in other words the elements are completely out of phase which results in severe degradation of the array performance. In order to reduce the errors associated with this differential spatial delay for flat reflectarrays, the integer N has to be reduced. Ideally, the error could be completely removed if true‐time‐delay lines can be used; that is, no phase wrap. However, as seen earlier, in a reflectarray this could correspond to several phase cycles, making it impractical. A more practical solution is to use elements that can provide more than a 2π cycle to alleviate this phase error, such as elements with longer delay lines. The main issues for such designs would be the available real estate in the lattice and losses in the delay lines that make the element design quite complicated. 5.1.3  Aperture Phase Error and Reflectarray Bandwidth Limitations To get a better understanding on the bandwidth performance of a reflectarray antenna, it is necessary to observe the phase error performance for all the elements with respect to a common reference point. Such an approach takes into account the relative phase errors of the elements as well as the effects of spatial phase delay. Theoretically, the relative phase requirement on an array surface can be calculated with respect to any element in the array. However, since in most reflectarray designs the feed is pointing to the geometrical center of the array, we consider this point as the reference for phase calculations. The general formula for an ideal phase relation is defined as





ψ i ( f ) −ψ ref ( f ) = k ⋅ { Ri − Rref − ( ri ⋅ rˆo − rref ⋅ rˆo )}. (5.9)

 Since the center element is used as the phase reference, rref 0. The relative phase th error for the i element in the array with respect to the center of the array can then be calculated as

Bandwidth of Reflectarray Antennas



 Phase Errori ( f ) = {ψ i ( f ) −ψ ref ( f )} − k ⋅ { Ri − Rref − ri ⋅ rˆo }. (5.10)

This definition of phase error on the array surface takes into account the relative phase requirements for every element of the array for a general array geometry and beam direction. To further expand this phase error formulation for a practical case, the feed illumination must also be taken into account. The reason is that the excitation of each element will determine the overall effect of the phase error of that individual element on the reflectarray performance. The illumination of a reflectarray antenna is calculated using the normalized radiation pattern of the feed horn simulated as a Cosq (θ) function, which is the widely adopted model to approximate practical feed patterns, as discussed in Chapter 2. The weighted phase error (WPE) is then defined as the product of the phase error and the normalized array illumination, that is,

WPE i f

Phase Errori f

Normalized Illumination i f . (5.11)

To quantitatively analyze the performance of phase errors, we consider a practical example of a Ka‐band reflectarray. A circular aperture reflectarray with a diameter of 159 mm is designed for the operating frequency of 32 GHz to generate a beam in the direction of (θ, φ) = (25°, 0°). The phase center of the feed horn is placed at X = −45.9 mm, Y = 0 mm, Z = 98.4 mm based on the coordinate system shown in Figure 5.6. For the horn model used in this study, the power q of the feed radiation pattern varies linearly from 5 at 30 GHz to 8.3 at 34 GHz. From the system design specifications, the location of all the array elements, the required element phase shift, and the illumination on the array surface can be calculated. The patch dimensions are then determined from the unit‐cell simulations at the center design frequency of 32 GHz. From these patch size dimensions, the frequency behavior of the elements reflection phase is obtained across the frequency band. The WPE on the array surface can then calculated at any specified frequency using (5.11), which is plotted in Figure 5.7 for two off‐center frequencies. Z

ˆro

⇀

θo

Ri

Y

⇀

ri

φo

i th element X

Figure 5.6  The coordinate system of the reflectarray antenna.

119

120

Reflectarray Antennas 70 60 50 15

40 30

10

50

20

0 15

5 10

10 xaxis (mm)

5

0

yaxis (mm)

(a)

70 60 50 15

40 30

10

50

20

0 15

5 10

xaxis (mm)

5

(b)

10 0

yaxis (mm)

Figure 5.7  Phase errors on the reflectarray aperture at off‐center frequencies: (a) 30 GHz, (b) 34 GHz.

The phase error plots on the reflectarray aperture provide a clear visual understanding of the bandwidth limitations in reflectarray systems. It should be noted that similar results were observed at different frequencies across the band. Also note that in these results, the quantization errors associated with the design are not taken into account. Although as mentioned earlier, the effect of quantization errors is not significant in these designs. To further aid in quantifying the bandwidth performance of reflectarrays, it is beneficial to study the system phase error, in other words, a single phase error number for any reflectarray systems at every frequency. Here we define the phase error of a reflectarray antenna system using two different definitions, namely

Average Phase Error f

1 N

N i 1

Phase Errori f , (5.12)

Bandwidth of Reflectarray Antennas Average Weighted Average

∣Phase Error∣ (degrees)

40 30 20 10 0 30

31

32

33

34

Frequency (GHz)

Figure 5.8  Average phase error of the reflectarray antenna as a function of frequency.



Average Weighted Phase Error f

1 N

N

WPEi f . (5.13)

i 1

Here, N is the number of phasing elements of the reflectarray antenna. Note that in comparison between the two equations, the latter also includes the amplitude taper of the reflectarray aperture. These phase error terms as a function of frequency are given in Figure  5.8 for the previous reflectarray system, where it can be seen that at off‐center frequencies phase errors will be generated on the aperture, and grow rapidly as we move away from the center frequency, ultimately resulting in bandwidth limitations of the reflectarray antenna. It is important to note that, while there is no direct correlation between these phase error terms and the gain bandwidth of a reflectarray, the formulation provided in this section makes it possible to quantify the bandwidth performance of different reflectarray elements and systems using basic element reflection coefficient data.

5.2 ­Reflectarray Element Bandwidth 5.2.1  Physics of Element Bandwidth Constraints Various designs of broadband elements have been demonstrated over the years [6]–[15]. As discussed earlier, regardless of the choice of phase tuning approaches, reflectarray elements exhibit some form of phase error as a function of frequency, which ultimately results in bandwidth reduction of the reflectarray system. Depending on the phase tuning approaches. however, the physics behind the bandwidth limitation is different. In the delay‐line approach, impedance matching between the transmission‐line and the patch is the key factor, since it ensures that the power is received or transmitted from the patch to the delay‐line and vice versa, thus controlling the element bandwidth. It is important to point out that while in principle, a good matching across the band will provide the appropriate phasing as a function of frequency, the resonances of the combined patch and line structure will result in somewhat non‐linear phase curves versus stub length. On the other hand, for the variable‐size approach, element bandwidth cannot be defined in terms of input impedance bandwidth. In fact, a solid definition of element bandwidth is not available for this approach. The classic single‐layer variable‐size patch

121

122

Reflectarray Antennas

has a highly non‐linear reflection phase curve as a function of patch width. These phase curves typically show a very steep slope near resonance and slow variations elsewhere. As a result, the reflection phase is highly sensitive to the frequency variations near resonance, which results in a significant phase change as a function of frequency, limiting the bandwidth of the reflectarray system. The general rule of thumb in designing variable‐size elements with good bandwidth performance is to achieve phase curves that have a flat slope as a function of patch size and also show remain parallel to each other at different frequencies. This is typically achieved by using thick substrates and multi‐resonance configurations, such as stacked patches or multiple rings. The element rotation technique, which is popularly used for circularly polarized reflectarrays, has different physics that limit the operation bandwidth. In this approach, the reflected phase of the co‐polarized field is dependent on the angle of rotation, and remains constant as a function of the frequency. Hence, a major concern is to maintain a good co‐polarized field level over a broad frequency band. Meanwhile, since the spatial phase delay is frequency‐dependent and the reflection phase is frequency‐independent, according to (5.2), phase error occurs as the frequency changes, which limits the bandwidth of reflectarray systems, especially for large‐size reflectarrays. 5.2.2  Parametric Studies on Element Bandwidth In the previous section the basic physics of bandwidth limitations were outlined for all three phase tuning approaches. The variable‐size element, however, is the most widely adopted method for reflectarray antennas, therefore in this section we will present comparative studies for several element topologies. In this study, the element bandwidth is defined by

BW

fu max fl max . (5.14) fu max fl max / 2

To illustrate this, in Figure  5.9(a) we show an example of phase curves obtained at different frequencies for a reflectarray element, which is designed at the center frequency of 13.5 GHz. By calculating the phase differences between other frequencies and center frequency, the results were obtained and shown in Figure 5.9(b). Those two frequencies at which the maximum absolute value of phase difference is 45o are referred to as upper and lower frequencies limits, indicated by fu max and f l max, respectively. Once the upper and lower frequencies are obtained, the element bandwidth is calculated from (5.14). Essentially, in this approach the element bandwidth is defined by the maximum phase difference, which in this case is considered to be 45°. This bandwidth definition is a measure on how fast the element phase varies with frequency. If an element phase varies quickly with frequency, a narrow element bandwidth is obtained; otherwise, a broad element bandwidth is obtained. It is worthwhile pointing out that this element bandwidth is not the same as the reflectarray bandwidth. Nevertheless, a broad element bandwidth usually leads to a broad reflectarray bandwidth. Using this definition, in this section we study the bandwidth performance of three representative variable‐size elements; namely, the square patch, square loop, and cross‐ dipole, which are shown in Figure 5.10. For each element geometry, the design variable “a” is used to change the reflected phase of the element. The square loop and cross‐dipole element have a secondary design variable “w,” which slightly affect the element performance and are typically used to tune

Bandwidth of Reflectarray Antennas 180 135

12.5 GHz 13 GHz 13.5 GHz 14 GHz 14.5 GHz

Phase (degree)

90 45 0 –45 –90 –135 –180

1

2

3

4

(a)

5

6

7

8

9

10

Length (mm) 90

Phase error (degree)

60 fu

30

45°

0 fc

–30

45° fl

–60 –90

1

(b)

2

3

4

6

5

7

8

9

10

Length (mm)

Figure 5.9  (a) Phase curves obtained at different frequencies. (b) The definition for the upper and lower frequency limits in the definition of element bandwidth.

w a

(b) w a

a

(a)

(c)

Figure 5.10  Geometrical models of variable‐size elements: (a) square patch, (b) square loop, and (c) cross‐dipole.

123

Reflectarray Antennas

the element. All the three elements studied here are single‐resonant, and can only provide a single‐phase cycle with a phase tuning range less than 2π. The elements are designed at the center frequency of 13.5 GHz. The unit‐cell size is 11.2 mm, and a substrate with a thickness of 1.58 mm and a dielectric constant of 2.55 is used for all elements. Figure 5.11 shows the phase curves of these three different elements at 13.5 GHz. The resonant dimensions, which correspond to a zero reflection phase, are around 5.65, 4.6, and 7.1 mm for patch, loop and cross‐dipole elements, respectively. Note that in these studies w was set to 0.5 mm for both square loop and cross‐dipole elements. The current distributions for the resonant dimensions of each element are also depicted in Figure 5.12, which shows that these three elements geometries resonate at the fundamental modes of patch, loop and dipole antennas. The element phase curves of these three elements at different frequencies are given in Figure 5.13. It can be seen that the patch element has the smoothest curve but with the smallest phase range [12]. The phase differences are also obtained with respect to the center frequency and are given in Figure  5.14, which show that the loop and 135 90 45 Phase (degree)

124

0 –45 –90

patch loop cross

–135 –180 –225

1

2

3

4

5

6

7

8

9

10

Length (mm)

Figure 5.11  Phase curves obtained at 13.5 GHz for three different elements. 5.65 mm

(a)

4.6 mm

(b)

7.1 mm

(c)

Figure 5.12  Current distributions at resonance dimensions for reflectarray elements: (a) patch, (b) loop, and (c) cross‐dipole.

Bandwidth of Reflectarray Antennas

180 135

Phase (degree)

90 45 0 12.5 GHz

–45

13 GHz

–90

13.5 GHz 14 GHz

–135 –180

14.5 GHz 1

2

3

4

5 6 7 Length (mm)

8

9

10

4

5 6 7 Length (mm)

8

9

10

4

6 7 5 Length (mm)

8

9

10

(a) 180 135

Phase (degree)

90 45 0 –45

12.5 GHz

–90

13 GHz 13.5 GHz

–135

14 GHz

–180 –225

14.5 GHz 1

2

3

(b) 180 135

Phase (degree)

90 45 0 –45

12.5 GHz

–90

13 GHz 13.5 GHz

–135

14 GHz

–180 –225

(c)

14.5 GHz 1

2

3

Figure 5.13  Reflection phase curves of reflectarray elements at different frequencies: (a) patch, (b) loop, and (c) cross‐dipole.

125

150 12.5 GHz 13 GHz 13.5 GHz 14 GHz 14.5 GHz

120

Phase (degree)

90 60 30 0 –30 –60 –90 –120 –150

1

2

3

4

(a)

5

6

7

8

9

10

Length (mm) 150 12.5 GHz 13 GHz 13.5 GHz 14 GHz 14.5 GHz

120 Phase error (degree)

90 60 30 0 –30 –60 –90 –120 –150

1

2

3

4

(b)

7

8

9

10

6 7 5 Length (mm)

8

9

10

5

6

Length (mm) 150 120

Phase error (degree)

90 60 30 0 –30

12.5 GHz

–60

13 GHz 13.5 GHz

–90

14 GHz

–120 –150

(c)

14.5 GHz 1

2

3

4

Figure 5.14  Phase errors compared to the one obtained at center frequency for: (a) patch (b) loop (c) cross‐dipole.

Bandwidth of Reflectarray Antennas

Table 5.1  Summary of reflectarray element performances. Element

Patch

Loop

Cross‐dipole

Phase range @ 13.5 GHz

293.0o

339.7o

335.6o

Element bandwidth

7.6%

5.3%

4.8%

cross‐dipole elements have much higher phase differences than the patch. The elements phase range at 13.5 GHz and element bandwidth as defined in (5.14) are summarized in Table 5.1, which shows that in terms of phase range loop outperforms the others, and in terms of bandwidth, the patch element provides the best performance. In the next stage, we study the effect of the substrate thickness on element performance. When the substrate thickness is increased, usually the reflection phase slope shows a more linear response at the cost of a reduced phase range, thereby essentially increasing the element bandwidth and decreasing the quantization error in fabrication. Theoretically, it could be explained by Q factor theory in patch antennas. It is well known that when the substrate thickness increases, the Q factor decreases, thus the element bandwidth is increased. In order to show the effects of the substrate thickness, the three elements are investigated with three different substrate thicknesses; namely 0.79, 1.58, and 3.18 mm. The phase curves obtained at 13.5 GHz for the patch elements with different substrate thicknesses are given in Figure 5.15, where it can be seen that as the substrate thickness increases, the phase curves become more linear and the phase range is greatly reduced. In order to study the element bandwidth, phase curves obtained at different frequencies are given in Figure  5.16. Phase differences are also computed and depicted in Figure 5.17. It is seen that for the thinnest laminate thickness of 0.79 mm, the phase error could reach around 90o when the frequency changes only by 0.5 GHz. On the other hand, for the thickest laminate thickness of 3.18 mm, a frequency shift of 1 GHz only results in a phase error around 30o. The phase ranges at 13.5 GHz and the element Figure 5.15  Comparison of phase curves obtained at 13.5 GHz for patch elements with different substrate thicknesses.

200

0.79 mm 1.58 mm 3.18 mm

Phase (degree)

100

0

–100

–200

0

2

4 6 Length (mm)

8

10

127

Reflectarray Antennas 180 135

Phase (degree)

90 45 12.5 GHz

0

13 GHz

–45

13.5 GHz

–90

14 GHz 14.5 GHz

–135 –180

1

2

3

4

(a)

5

6

7

8

9

10

Length (mm) 180 12.5 GHz

135

13 GHz 13.5 GHz

Phase (degree)

90

14 GHz

45

14.5 GHz

0 –45 –90 –135 –180

1

2

3

4

(b)

5

6

7

8

9

10

Length (mm)

180

12.5 GHz

135

13 GHz 13.5 GHz

90 Phase (degree)

128

14 GHz

45

14.5 GHz

0 –45 –90 –135 –180

(c)

1

2

3

4

5

6

7

8

9

10

Length (mm)

Figure 5.16  Comparison of phase curves for the patch elements with different substrate thicknesses: (a) t = 0.79 mm, (b) t = 1.58 mm, and (c) t = 3.18 mm.

Bandwidth of Reflectarray Antennas 150

12.5 GHz

120

13 GHz

Phase error (degree)

90

13.5 GHz

60

14 GHz

30

14.5 GHz

0 –30 –60 –90 –120 –150

1

2

3

4

(a) 150

8

9

10

5 6 7 Length (mm)

8

9

10

12.5 GHz

120

13 GHz

90 Phase (degree)

5 6 7 Length (mm)

13.5 GHz

60

14 GHz

30

14.5 GHz

0 –30 –60 –90 –120 –150

1

2

3

4

(b) 150

12.5 GHz

120

13 GHz

Phase error (degree)

90

13.5 GHz

60

14 GHz

30

14.5 GHz

0 –30 –60 –90 –120 –150

(c)

1

2

3

4

5 6 7 Length (mm)

8

9

10

Figure 5.17  Comparison of phase differences for patch elements with different substrate thicknesses: (a) t = 0.79 mm, (b) t = 1.58 mm, and (c) t = 3.18 mm.

129

Reflectarray Antennas

Table 5.2  Summary of phase ranges at 13.5 GHz and element bandwidth for the patch elements with different substrate thicknesses. Element

t = 0.79 mm

t = 1.58 mm

t = 3.18 mm

Phase range (@13.5 GHz)

327.6o

293.0o

191.9o

Element bandwidth

5.1%

7.6%

14.7%

bandwidth for the patch elements with various substrate thicknesses are summarized in Table 5.2. Similar analysis is performed to understand the element characteristics of the square loop and the cross‐dipole for different substrate thicknesses. These results are given in Figure 5.18 to Figure 5.23, and summarized in Tables 5.3 and 5.4. The observations are the same as the patch case: the thicker the substrate, the smaller the phase range and the wider the bandwidth. The results presented here clearly show that in all element geometries using a thick substrate is advantageous in terms of element bandwidth. However, as discussed in earlier chapters, a phase tuning range below 300° typically results in poor performance of the array, hence appropriate compromises have to be made when selecting the substrate thickness. As discussed earlier, the loop and cross‐dipole elements offer a second degree of freedom for the element design. The width of the trace in both element geometries can be used to fine‐tune the element performance. Here we study the effect of this design parameter on the performance of these elements by considering three different trace widths of 0.5, 1, and 1.5 mm. Similar analysis are conducted as before, and the results are given in Figure 5.24 to Figure 5.27, as well as Tables 5.5 and 5.6. The results presented here show that the change trace width slightly affects the bandwidth and phase range performance of both the loop and cross‐dipole elements. In the case of the loop, as the trace width increases, the bandwidth also increases by a small amount. Similar observations are made with the cross‐dipole element, and in general 200

Figure 5.18  Comparison of phase curves obtained at 13.5 GHz for the square loop elements with different substrate thicknesses.

0.79 mm 1.58 mm 3.18 mm

100 Phase (degree)

130

0 –100 –200 –300

0

2

4

6

Length (mm)

8

10

Bandwidth of Reflectarray Antennas 150

12.5 GHz

100

13 GHz 13.5 GHz

Phase (degree)

50

14 GHz

0

14.5 GHz

–50 –100 –150 –200 –250 1

2

3

4

(a)

5

6

7

8

9

10

Length (mm) 180 135

Phase (degree)

90

12.5 GHz

45

13 GHz 13.5 GHz

0

14 GHz

–45

14.5 GHz

–90 –135 –180 –225

1

2

3

4

5 6 7 Length (mm)

(b)

8

9

10

150

Phase (degree)

100 50

12.5 GHz 13 GHz

0

13.5 GHz

–50

14 GHz

–100

14.5 GHz

–150 –200 –250 1

(c)

2

3

4

5

6

7

8

9

10

Length (mm)

Figure 5.19  Comparison of phase curves for the square loop elements with different substrate thicknesses: (a) t = 0.79 mm, (b) t = 1.58 mm, and (c) t = 3.18 mm.

131

Reflectarray Antennas 150 12.5 GHz

120

13 GHz

Phase (degree)

90

13.5 GHz

60

14 GHz

30

14.5 GHz

0 –30 –60 –90 –120 –150

1

2

3

4

(a)

5

6

7

9

10

12.5 GHz

120

13 GHz

90 Phase error (degree)

8

Length (mm) 150

13.5 GHz

60

14 GHz

30

14.5 GHz

0 –30 –60 –90 –120 –150

1

2

3

4

(b)

5

6

7

8

9

10

Length (mm) 150

12.5 GHz

120

13 GHz

90 Phase error (degree)

132

13.5 GHz

60

14 GHz

30

14.5 GHz

0 –30 –60 –90 –120 –150

(c)

1

2

3

4

5

6

7

8

9

10

Length (mm)

Figure 5.20  Comparison of phase differences for the square loop elements with different substrate thicknesses: (a) t = 0.79 mm, (b) t = 1.58 mm, and (c) t = 3.18 mm.

Bandwidth of Reflectarray Antennas

Figure 5.21  Comparison of phase curves obtained at 13.5 GHz for the cross‐dipole elements with different substrate thicknesses.

200

Phase (degree)

100 0 –100 0.79 mm 1.58 mm

–200

3.18 mm –300

2

8

4 6 Length (mm)

10

150

Phase (degree)

100 50 0 –50

12.5 GHz

–100

13 GHz

–150

13.5 GHz

–200

14 GHz 14.5 GHz

–250 1

2

3

4

(a)

5

6

7

8

9

10

7

8

9

10

Length (mm) 150 100

Phase (degree)

Figure 5.22  Comparison of phase curves for the cross‐dipole elements with different substrate thicknesses: (a) t = 0.79 mm, (b) t = 1.58 mm, and (c) t = 3.18 mm.

0

50 0 12.5 GHz

–50

13 GHz

–100 –150

13.5 GHz 14 GHz

–200

14.5 GHz

–250 1

(b)

2

3

4

5

6

Length (mm)

133

Reflectarray Antennas

Figure 5.22  (Continued)

150

Phase (degree)

100 50 0 –50 –100

12.5 GHz 13 GHz

–150

13.5 GHz 14 GHz

–200

14.5 GHz

–250 1

2

3

4

(c)

6

7

8

9

10

Figure 5.23  Comparison of phase differences for the cross‐dipole elements with different substrate thicknesses: (a) t = 0.79 mm, (b) t = 1.58 mm, and (c) t = 3.18 mm.

12.5 GHz

120 Phase error (degree)

5

Length (mm)

150

13 GHz

90

13.5 GHz

60

14 GHz

30

14.5 GHz

0 –30 –60 –90 –120 –150

1

2

3

4

(a)

5

6

7

8

9

10

7

8

9

10

Length (mm) 150 120

Phase error (degree)

134

90 60 30 0 12.5 GHz

–30

13 GHz

–60

13.5 GHz

–90

14 GHz

–120 –150

(b)

14.5 GHz 1

2

3

4

5

6

Length (mm)

Bandwidth of Reflectarray Antennas

Figure 5.23  (Continued)

150

12.5 GHz

Phase error (degree)

120

13 GHz

90

13.5 GHz

60

14 GHz

30

14.5 GHz

0 –30 –60 –90 –120 –150

1

2

3

4

(c)

5

6

7

8

9

10

Length (mm)

Table 5.3  Summary of phase ranges at 13.5 GHz and element bandwidth for the square loop elements with different substrate thicknesses. Element

t = 0.79 mm

t = 1.58 mm

t = 3.18 mm

Phase range (@13.5 GHz)

353.7o

339.7o

281.1o

Element bandwidth

3.9%

5.3%

11.9%

Table 5.4  Summary of phase ranges at 13.5 GHz and element bandwidth for patch elements with different substrate thickness. Element

t = 0.79 mm

t = 1.58 mm

t = 3.18 mm

Phase range (@13.5 GHz)

352.2o

335.6o

268.2o

Element bandwidth

3.7%

4.8%

10.4%

for these types of configurations a larger trace width is advantageous to achieve a better performance.

5.3 ­Reflectarray System Bandwidth 5.3.1  Effect of Aperture Size on Reflectarray Bandwidth After the study on the bandwidth of reflectarray elements, now we focus on the bandwidth performance of reflectarray antennas. As discussed earlier, for large reflectarray antennas, the primary factor limiting the bandwidth is the effect of spatial phase delay. To quantitatively demonstrate the effect of aperture size on reflectarray bandwidth, here we consider four cases with aperture sizes of 13, 25, 50, and 100λ at the center design

135

Reflectarray Antennas 135 12.5 GHz

90

13 GHz

Phase (degree)

45

13.5 GHz 14 GHz

0

14.5 GHz

–45 –90 –135 –180 –225

1

2

3

4

(a)

5

6

7

8

9

10

Length (mm)

135 12.5 GHz

90

13 GHz 13.5 GHz

Phase (degree)

45

14 GHz

0

14.5 GHz

–45 –90 –135 –180 –225

1

2

3

4

(b)

6 7 5 Length (mm)

135

Phase (degree)

136

8

9

10

12.5 GHz

90

13 GHz

45

13.5 GHz 14 GHz

0

14.5 GHz

–45 –90 –135 –180 –225

(c)

1

2

3

4

6 7 5 Length (mm)

8

9

10

Figure 5.24  Comparison of phase curves for the square loop elements with trace width: (a) w = 0.5 mm, (b) w = 1 mm, and (c) w = 1.5 mm.

Bandwidth of Reflectarray Antennas 150

12.5 GHz

120

13 GHz

Phase error (degree)

90

13.5 GHz

60

14 GHz

30

14.5 GHz

0 –30 –60 –90 –120 –150

1

2

3

4

(a)

5

6

7

150

9

10

12.5 GHz 13 GHz

120 90 Phase error (degree)

8

Length (mm)

13.5 GHz

60

14 GHz 14.5 GHz

30 0 –30 –60 –90 –120 –150

1

2

3

4

(b)

5 6 7 Length (mm)

8

9

10

150 120 Phase error (degree)

90 60 30 0 12.5 GHz

–30

13 GHz

–60

13.5 GHz

–90

14 GHz

–120 –150

(c)

14.5 GHz 1

2

3

4

5

6

7

8

9

10

Length (mm)

Figure 5.25  Comparison of phase differences for the square loop elements with trace width: (a) w = 0.5 mm, (b) w = 1 mm, and (c) w = 1.5 mm.

137

Reflectarray Antennas 145 100

Phase (degree)

55 10 12.5 GHz

–35

13 GHz

–80

13.5 GHz

–125

14 GHz 14.5 GHz

–170 –215

1

2

3

4

(a)

5

6

7

8

9

10

6 7 5 Length (mm)

8

9

10

8

9

10

Length (mm) 145 100

Phase (degree)

55 10 –35

12.5 GHz

–80

13 GHz 13.5 GHz

–125

14 GHz

–170 –215

14.5 GHz 1

2

3

4

(b) 145 100 55 Phase (degree)

138

10 –35

12.5 GHz

–80

13 GHz 13.5 GHz

–125

14 GHz 14.5 GHz

–170 –215

(c)

1

2

3

4

5

6

7

Length (mm)

Figure 5.26  Comparison of phase curves for the cross‐dipole elements with trace width: (a) w = 0.5 mm, (b) w = 1 mm, and (c) w = 1.5 mm.

Bandwidth of Reflectarray Antennas 150 120 Phase error (degree)

90 60 30 0 12.5 GHz 13 GHz

–30 –60

13.5 GHz

–90

14 GHz

–120 –150

14.5 GHz 1

2

3

4

(a) 150

8

9

10

8

9

10

12.5 GHz

120

13 GHz

90 Phase error (degree)

5 6 7 Length (mm)

13.5 GHz

60

14 GHz

30

14.5 GHz

0 –30 –60 –90 –120 –150

1

2

3

4

(b) 150

6

7

12.5 GHz

120

13 GHz

90 Phase error (degree)

5

Length (mm)

13.5 GHz

60

14 GHz

30

14.5 GHz

0 –30 –60 –90 –120 –150

(c)

1

2

3

4

5

6

7

8

9

10

Length (mm)

Figure 5.27  Comparison of phase differences for the cross‐dipole elements with trace width: (a)  w = 0.5 mm, (b) w = 1 mm, and (c) w = 1.5 mm.

139

Reflectarray Antennas

Table 5.5  Summary of phase ranges at 13.5 GHz and element bandwidth for the square loop elements with different trace widths. Width (w)

0.5 mm

1 mm

1.5 mm

Phase range @13.5 GHz

339.7o

332.5o

327.0o

Element bandwidth

5.3%

6.0%

6.7%

Table 5.6  Summary of phase ranges at 13.5 GHz and element bandwidth for the cross‐dipole elements with different trace widths. Width (w)

0.5 mm

1 mm

1.5 mm

Phase range @13.5 GHz

335.6o

330.1o

324.3o

Element bandwidth

4.8%

5.3%

5.8%

50

Directivity (dB)

140

13λ0

45

25λ0

40

50λ0 100λ0

35 30 25 20 12

12.5

13

13.5

14

14.5

15

Frequency (GHz)

Figure 5.28  Directivity of the reflectarray antenna as a function of frequency for different aperture sizes.

frequency of 13.5 GHz. We consider the square loop element studied earlier in this chapter with a substrate thickness of 1.58 mm. For all configurations, the system f/D is set to 0.735. The reflectarray directivity is computed using the array theory approach described in Chapter 4 and is given in Figure 5.28. It can be seen that as the aperture size increases, the antenna directivity also increases; however, this is at the expense of a significant reduction in directivity bandwidth of the antenna. 5.3.2  Effects of Element on Reflectarray Bandwidth Although a quantitative relation between the element bandwidth and the reflectarray bandwidth is not available, a basic rule of thumb in reflectarray system design exists, which is if an element shows good bandwidth performance, it would translate to wide

Bandwidth of Reflectarray Antennas 180 135 Phase (degree)

90 45 0 –45 –90 –135

0.79 mm 1.58 mm 3.18 mm

–180 –225 –270 1

2

(a)

3

4

5

6

7

8

9

10

Length (mm) 36

Directivity (dB)

32

28 0.79 mm 1.58 mm 3.18 mm

24

20 12

(b)

12.5

13 13.5 14 Frequency (GHz)

14.5

15

Figure 5.29  Performance of a variable‐size square patch reflectarray using different substrate thickness: (a) Phase curves as a function of patch length, (b) directivity bandwidth of the reflectarray antenna with various thicknesses.

bandwidth performance for the reflectarray system. To quantify the effects of elements on the bandwidth of the reflectarray antenna, in this section we will study a reflectarray system using different types of elements. The reflectarray is designed for the center frequency of 13.5 GHz and has a diameter of 516 mm, with an f/D of 0.735. First, we consider the patch element and study the bandwidth performance of the reflectarray using different substrate thicknesses. Phase curves as a function of patch length and reflectarray directivity bandwidth are given in Figure 5.29. The directivity bandwidth results given here clearly show that as the substrate thickness increases, the reflectarray bandwidth increases. Note that, however, for the very thick substrate (3.18 mm), the significantly reduced phase range of 192° results in a reduction of directivity by about 1.5 dB. As mentioned earlier, increasing the substrate thickness to a point where the element phase range drops below the 300° range is not practical. A summary of these results is given in Table 5.7. It is worthwhile pointing out that the reflectarray bandwidth here is defined by 1 dB directivity drop.

141

Reflectarray Antennas

Table 5.7  Summary of directivity and bandwidth for reflectarray using the square patches. Substrate thickness

0.79 mm

1.58 mm

3.18 mm

Directivity @13.5GHz (dB)

35.7

35.7

34.2

Bandwidth

5.5%

10.6%

15.0%

Figure 5.30  Performance of a variable‐ size square loop reflectarray using different substrate thicknesses: (a) phase curves as a function of loop length, (b) directivity bandwidth of the reflectarray antenna with various thicknesses.

180 135 Phase (degree)

90 45 0 –45 –90 –135

0.79 mm 1.58 mm 3.18 mm

–180 –225 –270

1

2

3

4

(a)

6 7 5 Length (mm)

8

9

10

36

32 Directivity (dB)

142

28 0.79 mm 1.58 mm 3.18 mm

24

20 12

(b)

12.5

13

13.5

14

14.5

15

Frequency (GHz)

Similar studies are also conducted for reflectarrays consisted of the square loop and cross‐dipole elements, and results are given in Figure 5.30 and Figure 5.31, and summarized in Tables 5.8 and 5.9. The results presented here clearly show a distinct trend that bandwidth will improve by using thicker substrates regardless of the element geometry. One thing that needs to be pointed out is that for the cross‐dipole configuration when the substrate thickness is 1.58 mm, the directivity is highest. The reason is that there are two main factors controlling the element phase errors at the design frequency, namely, the phase range and the quantization error, which ultimately influence the peak directivity. When the

Bandwidth of Reflectarray Antennas

Figure 5.31  Performance of a variable‐ size cross‐dipole reflectarray using different substrate thicknesses: (a) phase curves as a function of dipole length, (b) directivity bandwidth of the reflectarray antenna with various thicknesses.

180 135 Phase (degree)

90 45 0 –45 –90 –135

0.79 mm 1.58 mm 3.18 mm

–180 –225 –270

1

2

4

3

(a)

5 6 7 Length (mm)

8

9

10

36

Directivity (dB)

32

28 0.79 mm

24

20 12

1.58 mm 3.18 mm 12.5

(b)

13 13.5 14 Frequency (GHz)

14.5

15

Table 5.8  Summary of directivity and bandwidth for reflectarray using the square loops. Substrate thickness

0.79 mm

1.58 mm

3.18 mm

Directivity @13.5GHz (dB)

35.5

35.7

35.6

Bandwidth

0.2%

6.2%

14.8%

Table 5.9  Summary of directivity and bandwidth for reflectarray using the cross‐dipoles. Substrate thickness

0.79 mm

1.58 mm

3.18 mm

Directivity @13.5GHz (dB)

35.6

35.7

35.6

Bandwidth

0.2%

4.5%

14.0%

substrate thickness differs, both phase range and quantization error vary, which changes the phase errors in two opposite ways. When the substrate thickness is 1.58 mm, it allows a smaller quantization error compared to thinner substrate, while a larger phase range compared to the thicker one; hence it has the smallest total phase error and consequently the largest peak directivity.

143

Reflectarray Antennas

Figure 5.32  Directivity as a function of frequency for reflectarrays using different elements.

36

32 Directivity (dB)

144

28

patch loop cross

24

20 12

12.5

13

13.5

14

14.5

15

Frequency (GHz)

As a final remark on this comparative study, it is interesting to compare the performance of these reflectarrays using the three different element shapes. A substrate thickness of 1.58 mm is used for all three designs and the aperture size is 25λ at the center design frequency of 13.5 GHz. The directivity bandwidths are given in Figure 5.32 where it can be seen that the patch element performs best. The 1‐dB directivity bandwidth for these reflectarrays are 10.6, 6.2, and 4.5% for the patch, square loop, and cross‐dipole, respectively. It is worthwhile pointing out that the reflectarray bandwidth results here has the same trend as the element bandwidth results in Table 5.1. In summary, different methodologies have been proposed to design reflectarray antennas with a wide bandwidth. Typically, a thick substrate has to be utilized and multiple resonances need to be introduced to provide a good phase tuning range with a linear slope. Discussions on wide band reflectarray antennas will be given in Chapter 7.

­References 1 C. J. Sletten, Reflector and Lens Antennas: Analysis and Design Using Personal Computers,

Artech House Inc., 1988.

2 J. Huang and J. A. Encinar, Reflectarray Antennas. New York, NY, USA: Wiley‐IEEE, 2008. D. M. Pozar, S. D. Targonski, and H. D. Syrigos, “Design of millimeter wave microstrip 3

reflectarrays,” IEEE Trans. Antennas Propag., Vol. 45, No. 2, pp. 287–296, Feb. 1997.

4 J. Huang, “Bandwidth study of microstrip reflectarray and a novel phased reflectarray

5 6 7 8

concept,” presented at the IEEE Antennas and Propagation Society International Symposium, California, USA, June 1995. D. M. Pozar, “Bandwidth of reflectarrays,” Electron. Lett., Vol. 39, No. 21, Oct. 2003. J. A. Encinar, “Design of two‐layer printed reflectarrays using patches of variable size,” IEEE Trans. Antennas Propag., Vol. 49, No. 10, pp. 1403–1410, Oct. 2001. N. Misran, R. Cahill, and V. Fusco, “Reflection phase response of microstrip stacked ring elements,” Electron. Lett., Volume 38, No. 8, pp. 356–357, April 2002. J. A. Encinar and J. A. Zornoza, “Broadband design of three‐layer printed reflectarrays,” IEEE Trans. Antennas Propag., Vol. 51, No. 7, pp. 1662–1664, July 2003.

Bandwidth of Reflectarray Antennas

9 M. E. Bialkowski and K. H. Sayidmarie, “Investigations into phase characteristics of a

10

11

12

13

14

15

single‐layer reflectarray employing patch or ring elements of variable size,” IEEE Trans. Antennas Propag., Vol. 56, No. 11, pp. 3366–3372, Nov. 2008. M. Bozzi, S. Germani, and L. Perregrini, “Performance comparison of different element shapes used in printed reflectarrays,” in Proc. IEEE Antennas Wireless Propag. Lett., 2003, Vol. 2, pp. 219–222. N. Misran, R. Cahill, and V. Fusco, “Design optimization of ring elements for broadband reflectarray antennas,” Microwaves, Antennas and Propagation, IEE Proceedings, Vol. 150, No. 6, pp. 440–444, Dec. 2003. K. H. Sayidmarie and M. E. Bialkowski, “Investigations into unit cells offering an increased phasing range for single‐layer printed reflectarrays”, Microwave and Optical Technology Letters, Vol. 50, No. 4, pp. 1028–1032, Apr 2008. P. Nayeri, M. E. Bialkowski, and K. H. Sayidmarie, “Design of unit cells of double circular rings for a single‐layer microstrip reflectarray,” 20th Asia Pacific Microwave Conference, Hong Kong, China, Dec. 2008. M. R. Chaharmir, J. Shaker, M. Cuhaci, and A. Ittipiboon, “A broadband reflectarray antenna with double square rings,” Microw. Opt. Technol. Lett., Vol. 48, No. 7, pp.  1317–1320, Jul. 2006. J. Shaker, M. R. Chaharmir, and J. Ethier, Reflectarray Antennas, Analysis, Design, Fabrication, and Measurement, Artech House, 2013.

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147

6 Reflectarray Design Examples In the previous chapters, a systematic design and analysis procedure for reflectarray antennas was outlined in detail. In this chapter, we present several design examples of reflectarray antennas that can serve as a valuable reference for the readers. The first example is a comprehensive design of Ku‐band reflectarray, outlining all steps in design, analysis, fabrication, and testing. Moreover, several representative design examples, including a Ka‐band circularly polarized reflectarray antenna and a comparative study of Ku‐band reflectarrays using different element shapes, are also provided in a concise fashion.

6.1 ­A Ku‐band Reflectarray Antenna: A Step‐by‐Step Design Example 6.1.1  Feed Antenna Characteristics As discussed earlier, the feed antenna radiation characteristics are one of the most influential measures in the design. Similar to reflector antennas [1]–[3], typically the engineer has two options for the feed design. Either a specific feed antenna is to be used for the design, in which case the system parameters need to be selected appropriately; or a feed antenna can be designed that achieves optimum radiation characteristics for a desired system. While the latter option generally provides the engineer with several degrees of freedom in designing the reflectarray, low‐cost prototyping typically limits the engineer to the former option. For this Ku‐band design, we select a commercially available pyramidal horn antenna (LB 62–15) from A‐INFO [4]. The aperture opening of the horn is 50 × 35 mm2, and the antenna has a gain of 15.8 dBi at the center design frequency of 14.25 GHz. A photo of the horn antenna is given in Figure 6.1. To analyze the radiation characteristics, the antenna is modeled in the commercial EM solver, ANSYS HFSS [5]. The simulation model is shown in Figure 6.2. One of the notable advantages of the full‐wave simulation here is that in addition to calculating the far‐field radiation performance, it can also provide the electromagnetic field quantities inside the aperture of the horn which gives one a better understanding of the antenna radiation characteristics. The magnitudes of the total electric fields in E‐ and H‐planes Reflectarray Antennas: Theory, Designs, and Applications, First Edition. Payam Nayeri, Fan Yang, and Atef Z. Elsherbeni. © 2018 John Wiley & Sons Ltd. Published 2018 by John Wiley & Sons Ltd.

148

Reflectarray Antennas

Figure 6.1  The LB 62–15 pyramidal horn antenna from A‐INFO. Figure 6.2  The feed horn antenna model in ANSYS HFSS.

Z

X

0

Y

25

50 (mm)

at 14.25 GHz are shown in Figure 6.3. Note that in this setup, E‐ and H‐planes correspond to xz‐ and yz‐planes, respectively. It can be seen that the field distributions inside the horn antenna clearly show the transition from guided to free space waves. The horn is a classic aperture antenna, and as such, its radiation pattern is a manifest of the field distribution on its aperture. The tangential electric fields on the aperture are given in Figure 6.4, where it can be seen that, as expected for a pyramidal horn, the co‐ polarized (Ex) amplitude taper is different in the two orthogonal directions. Due to the

Reflectarray Design Examples

difference in both size and taper of the rectangular shaped aperture, the radiation pattern of a pyramidal horn will exhibit some asymmetry [6], [7]. As shown in Figure 6.5, the horn radiates a directive beam, which is quite suitable for illuminating a reflectarray aperture. The simulated peak gain of the antenna is 15.74 dBi, which is very close to the value reported in the data sheet [4]. The half‐power beam‐width in both E‐ and H‐planes is about 30 degrees, and the side‐lobe level is −14 dB. As discussed earlier, the pattern of the feed antenna is slightly asymmetric; and while for a reflectarray a feed antenna with a symmetric radiation pattern is usually more desirable, the difference in this feed horn is quite small. In fact, the 2D radiation patterns given in Figure 6.6 show that the patterns in both planes match up very closely up to about ±30°, which is quite sufficient for a reflectarray antenna design. As discussed in Chapter 3, it is desirable to use a radiation pattern model for the feed antenna in order to effectively design and analyze the reflectarray system. For this design, we use the cosine‐q radiation pattern model for the feed antenna. The simulated radiation pattern of the antenna is therefore matched to a cosine‐q model in order to z

E Field [V/ m] 1. 0000E+004 9. 3333E+003 8. 6667E+003 8. 0000E+003 7. 3333E+003 6. 6667E+003 6. 0000E+003 5. 3333E+003 4. 6667E+003 4. 0000E+003 3. 3333E+003 2. 6667E+003 2. 0000E+003 1. 3333E+003 6. 6667E+002 0.0000E+000

(a)

x

0

25

50 (mm)

Figure 6.3  The electric field magnitude inside the horn antenna at 14.25 GHz: (a) E‐plane, (b) H‐plane.

149

150

Reflectarray Antennas

z

E Field [V / m] 1. 0000E+004 9. 3333E+003 8. 6667E+003 8. 0000E+003 7. 3333E+003 6. 6667E+003 6. 0000E+003 5. 3333E+003 4. 6667E+003 4. 0000E+003 3. 3333E+003 2. 6667E+003 2. 0000E+003 1. 3333E+003 6. 6667E+002 0.0000E+000

(b)

x

0

25

50 (mm)

Figure 6.3  (Continued)

determine the value of q. For this horn the value of q is 10.5, which matches the simulated patterns very well up to about ±30°. Within this angular range, the phase center of this horn is 10.16 mm below the surface of the horn aperture. 6.1.2  Reflectarray System Design With the feed antenna selected, the next stage is to design the reflectarray system. As discussed in earlier chapters, the antenna gain relates directly to the aperture size, and as such selecting the appropriate aperture size is typically the first step in system design. In order to achieve a moderately high gain with this reflectarray, we select a circular aperture with a diameter of 360 mm, which is about 17 wavelengths at the center design frequency of 14.25 GHz, and can provide us with a theoretical maximum aperture directivity of 34.6 dBi. Feed antenna placement is the most important and influential step in the reflectarray system design. To avoid feed blockage, we design an offset system with a feed offset angle of 15°, where the feed points to the geometrical center of the array. The optimum

1500

1500 20

10

1000

0 500

–10

y-axis (mm)

y-axis (mm)

20

–20

1000

0 500

–10 –20

–10

(a)

10

0

10

0

–10

(b)

x-axis (mm)

0

10

x-axis (mm)

20

20 10 0

0 –10

100 y-axis (mm)

y-axis (mm)

100

–100

–20 –10

(c)

0

0

10 0

0 –10

–100

–20

10

–10

(d)

x-axis (mm)

0

10

x-axis (mm)

Figure 6.4  Electric fields on the aperture of the horn antenna: (a) |Ex| (V/m), (b) |Ey| (V/m), (c) phase of Ex (degrees), and (d) phase of Ey (degrees).

Normalized Gain Pattern (dB)

Figure 6.5  A 3D pattern of the horn antenna.

0 –5 –10 –15 –20

E-plane H-plane cos10.5 θ model

–25 –30 –60

–40

–20

0

20

40

60

θ (degrees)

Figure 6.6  Normalized and cosine‐q model of the radiation patterns of the horn antenna.

Reflectarray Antennas

position of the feed is then determined by means of aperture efficiency analysis. The reflectarray efficiency is computed using the formulation given in Chapter 3, and efficiency as a function of focal point to aperture distance (HF) is given in Figure 6.7. The peak aperture efficiency for this system is about 78.15% when HF equals 370 mm. However, as discussed earlier in Chapter  3, it is also important to consider the edge taper on the aperture. Typically, a slight reduction of peak aperture efficiency will result in a better edge taper for a reflectarray. After some parametric studies, we select HF to be 342.9 mm, that is the feed phase center is placed at x = −91.88 mm, y = 0, and z = 342.90 mm. This corresponds to an aperture efficiency of 78.10%, and an edge taper which is almost below −10 dB over most of the reflectarray edge. Also, it is important to point out that the maximum edge tapers for HF equal to 370 mm (peak efficiency) and 342.9 mm are −8.57 and −9.65 dB, respectively. Note that since the system is offset, the edge taper is not constant over the reflectarray edge. The illumination on the aperture and edge taper as a function of aperture azimuth angle are given in Figure 6.8. 100

Efficiency (%)

90 80 70 60

ηi ηs

50

ηa

40 200

250

300

350

400

450

500

HF

Figure 6.7  Reflectarray aperture efficiency as a function of focal point to aperture distance.

–100

–5

0

–10

100

–15 –100

(a)

–9 0

0

100

x-axis (mm)

Edge Taper (dB)

Illumination (dB)

y-axis (mm)

152

–10 –11 –12 –13

–20

(b)

0

50

100

150

ϕaperture (degrees)

Figure 6.8  (a) Illumination on the reflectarray aperture. (b) Edge taper as a function of aperture azimuth angle.

Reflectarray Design Examples

Table 6.1  Design parameters for the offset reflectarray and reflector. Reflectarray Reflector

DRA

HF

ΔB

θoffset

360 mm

342.9 mm

0 mm

15°

D

F

H

ψB

150.3 mm

343.44 mm

−12.45 mm

28.32°

350 300

x (mm)

250 200

DRA

150 θ offset

100 50

HF

0 –400

–300

–200

–100

0

z (mm)

Figure 6.9  Geometrical setup of the reflectarray and reflector in the reflector coordinate system.

As discussed in earlier chapters, the reflectarray systems designed in this section, along with the analogous reflector, can each be completely defined with four design parameters. These parameters are given in Table  6.1. Note that with this setup the reflector offset height is a negative number, which would cause some blockage for such a reflector system. However, one of the advantages of reflectarrays over reflectors is that the direction of the main beam can be further controlled by adding a progressive phase on the aperture. In this design, we select a main‐beam direction of 15° for the reflectarray, which will ensure that no feed blockage will be observed for the system and will also minimize beam squint for the reflectarray [8]. The geometrical setup of the reflectarray and the analogous reflector antenna is given in Figure 6.9. 6.1.3  Reflectarray Element Design A critical step in designing the reflectarray antenna aperture, is designing appropriate phasing elements. These elements need to provide a phase shift in order to compensate for the spatial delay of the feed antenna, as well as the necessary progressive phase to generate a collimated beam in the required direction. This reflectarray is designed to focus the main beam 15° off broadside. Using the formulation presented in Chapter 2, and the selected position for the feed antenna, the required phase shift on the aperture can be computed. An important consideration here is the size of the unit‐cells. Here we select a square lattice with a side dimension of 10 mm, which is 0.475 half‐wavelength at the

153

Reflectarray Antennas

center design frequency. With a diameter of 360 mm for the circular aperture of the reflectarray, this corresponds to 936 elements on the aperture. An image of the grid layout for the array and the required phase shift on the aperture are both given in Figure 6.10. Now that we have determined the required phase shift on the aperture, we need to design a phasing element that can provide this phase distribution on the aperture. For this reflectarray we select the variable size technique and use a classic square patch element. A 31 mil Taconic TLX8 laminate (εr = 2.55, tan δ = 0.0018) is selected for the substrate. The thickness is about 0.06 wavelengths in the dielectric, which should provide a good phase tuning range for the element. The element unit‐cell is modeled in Ansoft Designer [9], using the periodic boundary conditions. A snap shot of the simulation model is given in Figure 6.11.

y-axis [element number]

154

(a)

30 100 20

0

10

(b)

–100 10 20 30 x-axis [element number]

Figure 6.10  (a) Grid layout of the array. (b) Required phase shift on the reflectarray aperture.

Figure 6.11  Reflectarray element simulation model.

Reflectarray Design Examples

150

⦟Γ (deg.)

100 50 0 –50 –100 –150 2

(a)

4

6

8

10

Patch Width (mm) 1 0.99

ǀΓǀ

0.98 0.97 0.96 0.95

(b)

2

4 6 Patch Width (mm)

8

10

Figure 6.12  Reflection coefficients of the element at 14.25 GHz: (a) phase, (b) magnitude.

The reflection coefficients of the element as a function of patch width are given in Figure 6.12. This element provides a phase tuning range of 326°, which is quite sufficient for a reflectarray antenna. In addition, these elements provide a reflection magnitude that is close to unity, thus almost all the power impinging on the reflectarray aperture will be reflected. Note that due to fabrication considerations, the minimum and maximum sizes of the patches are selected to be 0.5 and 9.9 mm, respectively. While as discussed earlier, the phase variation obtained with these elements is non‐linear, with a fabrication precision of 0.05 mm (about 2 mils), the maximum phase quantization errors will be about 15°, which is acceptable for this relatively large array. The phasing elements designed here provide the means to realize the desired reflection phase on the aperture of the reflectarray antenna. Using the data in Figure 6.12(a), we select patch widths that provide reflection phases that match the ideal required phase distribution of Figure 6.10(b), to the best extent possible. The mask model of the reflectarray antenna is shown in Figure 6.13. It should be noted here that the elements are designed under normal incidence approximation, however, as discussed earlier; this approximation is quite valid for microstrip elements with thin substrates. Due to fabrication tolerances in the design, and the limited phase range of the elements that are less than the full phase cycle, the reflection phase of the reflectarray elements will not exactly match that of Figure 6.10(b). However, in our design, these

155

Reflectarray Antennas

Figure 6.13  Mask plot of the reflectarray antenna.

(a)

30 100 20

0

10

–100

10 20 30 X-axis [element number]

Y-axis [element number]

Y-axis [element number]

156

(b)

15

30

10

20

5

10

10 20 30 X-axis [element number]

0

Figure 6.14  (a) Quantized phase distribution on the aperture. (b) Quantization phase errors on the aperture.

errors are relatively small. The quantized phase distribution on the aperture, as well as the relative phase errors are given in Figure 6.14, where it can be seen that the aperture phase distribution obtained by these elements is very close to the ideal case. 6.1.4  Radiation Analysis With the reflectarray mask designed in the previous section, we have completed all the design stages of the reflectarray antenna. The next important task is to verify the design by analyzing the radiation characteristics of the antenna. In this section, we compute the radiation pattern of the antenna using two approaches: the array theory approach and the full‐wave simulation of the entire reflectarray. We start with the robust array theory analysis technique described in Chapter 4. This method of analysis is very fast and can help us get a good idea about the radiation characteristics of the antenna. Two‐dimensional plane cuts of the radiation pattern of the reflectarray

Reflectarray Design Examples

Radiation Pattern (dB)

0

–20

–40

–60

–50

(a)

0

50

α1 (degress)

Radiation Pattern (dB)

0

–20

–40

–60

(b)

–50

0 α2 (degrees)

50

Figure 6.15  Radiation pattern of the reflectarray antenna in the principal planes: (a) P.P.1, (b) P.P.2.

in its principal planes at 14.25 GHz are given in Figure 6.15, where a high gain pencil beam radiation pattern is clearly observed. Also note that the beam is correctly scanned to 15° in the elevation plane (P.P.1). Note that these radiation patterns are obtained using ideal elements. In the next stage of the analysis, we now compute the radiation pattern of the antenna using the realistic phase shift, which are obtained by the patch elements. Note that in this case, limited phase range of the patch elements, along with the quantization effects of the elements phase, leads to some errors in the antenna performance. Comparison between the radiation patterns of the antenna using ideal elements and practical patch elements is given in Figure 6.16, where it can be seen that while the phase errors give rise to some degradation in antenna pattern, the effects are almost negligible. The radiation pattern of the antenna in the uv‐plane is also given in Figure  6.17, where it can be seen that the reflectarray shows a very directive pattern in the entire 3D space. In the final stage of radiation analysis, we compute the radiation pattern of the reflectarray antenna with a full‐wave electromagnetic simulation software. This method of analysis is highly accurate since it takes into account all approximations in the earlier approaches, however, the cost is a significantly high computational time and large memory requirements. For this simulation, the total computational time was 60 hours

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Figure 6.16  Comparison of the radiation pattern of the reflectarray antenna at 14.25 GHz using ideal and patch elements: (a) P.P.1, (b) P.P.2.

and 40 minutes, on an Intel Xeon E5–2680 processor, and the simulation utilized 141.1 GB of memory. Nonetheless, if resources are available, one generally would use this approach before prototyping the reflectarray. For our design, we simulated the antenna in ANSYS HFSS. Images of the simulation setup are given in Figure  6.18. The 3D radiation pattern of the reflectarray antenna is given in Figure 6.19, where it can be seen that a pencil beam radiation pattern is obtained with this reflectarray antenna as expected. It is also interesting to compare the radiation pattern obtained by the full‐wave simulation with the array theory approach. These are given in Figure 6.20, where it can be seen that a good agreement is observed between these approaches, in particular in the area of the main beam and the first side lobes. Outside the main‐beam area generally the full‐wave approach is considered to be more accurate. It is also important to note that the directivity computed using the array theory approach and the one obtained using HFSS are 33.49 dB and 32.21 dB, respectively. This 1.28 dB discrepancy in directivity is partially attributed to the approximations in the array theory and aperture efficiency analysis as described in Chapters 3 and 4.

Reflectarray Design Examples 1 –5 –10

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Figure 6.17  Radiation pattern of the reflectarray at 14.25 GHz in the uv‐plane: (a) contour plot, (b) 3D pattern.

6.1.5  Fabrication and Measurements In the final stage of the design process, the reflectarray antenna is fabricated and tested. Measurements are conducted using a planar NSI near‐field scanner. A photo of the test setup is given in Figure 6.21. The waveguide probe is an OEWG WR62, which operates from 12.4 to 18 GHz. The distance between the waveguide probe and the reflectarray aperture is 24 inches. The sampling plane dimension is 30.0 × 30.0 inch2 and the sample spacing is set to 0.25 inch. The average amplitude taper at the edge of the sampling plane is below 20 dB. Note that with the planar near‐field system, far‐field results are only valid within a certain observation angle, which is determined based on the size of the sampling plane, and the distance between the antenna under test (AUT) and the sampling plane. For this test setup, the maximum far‐field angle was set to ±45°. The sampled near‐field electric fields from the reflectarray antenna are given in Figure 6.22. Note that the horn antenna is x‐polarized, hence the y‐component of the sampled electric field has a significantly smaller magnitude. The sampled

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Figure 6.18  Images of the reflectarray antenna in ANSYS HFSS: (a) top view without the feed, (b) 3D view with the feed horn.

(a)

(b)

x‐component of the electric field shows a strong field at the center of the sampling plane as expected. More importantly note that in the area of the strong illumination, the phases of these fields also form parallel lines indicating a progressive phase shift to generate a titled beam.

Reflectarray Design Examples Z

dB(GainTotal) 3.2119E + 001 2.8644E + 001 2.5170E + 001 2.1695E + 001 1.8221E + 001 1.4746E + 001 1.1271E + 001 7.7968E + 000 4.3222E + 000 8.4760E – 001 –2.6270E + 000 –6.1016E + 000 –9.5762E + 000 –1.3051E + 001 –1.6525E+001 –2.0000E+001

X

Theta

Ph1

Y

Figure 6.19  A 3D radiation pattern of the reflectarray antenna at 14.25 GHz obtained using ANSYS HFSS.

From these sampled near‐fields, the far‐field radiation pattern of the reflectarray antenna is obtained, which is given in Figure 6.23. It can be seen that these results match closely with the simulated results in Figure 6.17. The measured radiation patterns obtained here are also compared with the results of the full‐wave simulations in Figure 6.24, where it can be seen that a good agreement is observed. It is worthwhile to note that the discrepancies outside the area of the main beam is attributed to the fact that measured results obtained using planar near‐field systems tend to be less accurate outside a nominal 60° angular distance from the main beam [10]. Nonetheless, a very good agreement is observed in the main‐beam area. Also note that cross‐polarization is below 33 dB in the two principal planes. The measured gain and aperture efficiency of the antenna as a function of frequency are also given in Figure 6.25. The peak gain and aperture efficiency of this reflectarray are 32.28 dB and 59.8%, which are obtained at 14.1 GHz. At the center design frequency of 14.25 GHz, the measured gain is 32.06 dB, which is quite close

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Figure 6.20  Comparison of the radiation patterns of the reflectarray antenna at 14.25 GHz obtained using two different approaches: (a) P.P.1, (b) P.P.2.

Figure 6.21  Photo of the reflectarray antenna under test.

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Figure 6.22  Electric fields sampled by the near‐field scanner at 14.25 GHz: (a) |Ex| in dB, (b) phase of Ex in degrees, (c) |Ey| in dB, (d) phase of Ey in degrees.

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Figure 6.23  Measured radiation pattern of the reflectarray at 14.25 GHz in the uv‐plane: (a) contour plot, (b) 3D pattern.

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Figure 6.24  Comparison between the measured and simulated radiation pattern of the reflectarray antenna at 14.25 GHz: (a) P.P.1, (b) P.P.2.

Co-pol (Simulated) Co-pol (Measured) X-pol (Measured)

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Figure 6.25  Measured performance of the reflectarray antenna as a function of frequency: (a) gain, (b) aperture efficiency.

1 dB 3 dB

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Reflectarray Design Examples

to the value of 32.12 dB obtained using full‐wave simulations. The 1 dB gain bandwidth of the reflectarray (13.84–14.55 GHz) is about 5%, while the 3‐dB gain bandwidth 13.42–15.21 GHz is about 12.5%. The relatively narrow bandwidth of this reflectarray antenna is primarily attributed to the thin substrate used for the reflectarray elements.

6.2 ­A Circularly Polarized Reflectarray Antenna using an Element Rotation Technique In this section, we present a design for a circularly polarized reflectarray antenna using the single split ring element described in Chapter 2. The feed antenna is a circularly polarized corrugated conical horn antenna. The antenna has two ports allowing for both right‐hand and left‐hand circular polarizations, where the septum polarizer switch controls the polarization of the antenna [11]. For this design, the polarization of the feed antenna is set to be left‐handed. A photo of this horn antenna is given in Figure 6.26. As discussed earlier, the common approach to characterize the radiation performance of a feed horn antenna is to use a cosine‐q radiation pattern model, where the value of the parameter q is determined from the horn measurements. The measured radiation pattern of the horn antenna at 32 GHz is given in Figure 6.27, where it is also compared with a cosine‐q model with q = 6.5. It can be seen that, in both planes, the cosine‐q model gives a good representation of the horn pattern. The geometrical model of the single split ring (SSR) element labeled with design parameters is shown in Figure 6.28. The element is designed for 32 GHz operation. A 31 mil Rogers 5880 laminate is used for the element. The design parameters are Rin = 1.6 mm, Rout = 1.8 mm, and θs = 118°. The magnitudes of the reflected co‐polarized (LHCP) and the cross‐polarized (RHCP) components, with a normal incident plane wave source, are depicted in Figure 6.29(a). The reflection phase of the co‐polarized component as a function of rotation angle was

Figure 6.26  Photograph of the Ka‐band feed horn antenna.

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Figure 6.27  Measured far‐field patterns of the Ka‐band circularly polarized horn antenna.

Figure 6.28  The SRR element along with the design parameters. θs

Rin

Rout

given in Chapter 2, and thus not repeated here. As discussed earlier, the cross‐polarization level may increase to a large value when the incident angle θ grows up to 50° [12], as shown in Figure 6.29(b). When a prototype is built, the number of elements under an extreme incident angle is usually limited due to the system configuration. Hence this effect will be more or less mitigated. However, one may still foresee a relatively high cross‐polarization. The array is designed to achieve a nominal gain of 30 dB. A circular aperture is considered for the array with a diameter of 134 mm. This value has been extended to 138.2 mm for actual fabrication to include the elements on the boundary. An offset feed of 25° is applied due to its small blockage loss and convenience for measurement. The feed points to the geometrical center of the array, and the elements are rotated to provide a progressive phase that directs the reflectarray main beam in the specular direction, that is, 25° off broadside. To achieve maximum aperture efficiency, H, the feed position distance from the feed to the array plane is set to 98 mm. This corresponds to an H/D ratio of 0.74 and an aperture efficiency of 75%. The array has a total of 640 elements contained on the aperture. Photos of the array prototype and the reflectarray system are given in Figure 6.30.

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Figure 6.29  Magnitudes of co‐ and cross‐polarized element reflection coefficients as a function of frequency. (b) Effects of incident angle on the reflection coefficient magnitudes as a function of frequency.

(a)

(b)

Figure 6.30  Fabricated circularly polarized reflectarray antenna: (a) array prototype, (b) reflectarray system.

Reflectarray Antennas

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Figure 6.31  Comparison between measured and simulated radiation patterns of the reflectarray antenna at 32 GHz: (a) P.P.1, (b) P.P.2.

The radiation pattern of the antenna is computed using the aperture field method, and the antenna is tested using the planar near‐field measurement system. Comparisons between the measured and simulated results are given in Figure 6.31. The results agree well with each other in the main lobe region. Deviations in the side lobes below −20 dB are mainly due to the diffraction of the incidence at the edge of the finite ground plane. The measured main beam is found at 24.2°. The slight deviation to the designed value of 25° could be due to the error of alignment in the measurement setup. The side‐lobe levels are below −20 dB in both planes. The 3 dB beam widths are 4.6° and 5.2° in two principal planes, respectively. A relatively large cross‐polarization is observed which is primarily attributed to the high cross‐polarization of the feed antenna.

Reflectarray Design Examples

6.3 ­Bandwidth Comparison of Reflectarray Designs using Different Elements As discussed in the previous chapters, different phasing elements have been proposed for reflectarray antennas. While they all provide the necessary phase shift needed to collimate the beam, they do exhibit different characteristics. In this section, we will compare the effect of different geometries on the reflectarray performance, in particular, the bandwidth characteristic. To quantify some of these differences, in this section, three different reflectarray antennas, using square patch, cross dipole, and square loop elements, are compared. The geometrical models of the three different elements are depicted in Figure  6.32. The system setup for the measurement is also given in Figure 6.33. The horn antenna

w a

(a)

(b)

a

w

a

(c) Figure 6.32  Geometrical models of variable size elements: (a) square patch, (b) square loop, and (c) cross dipole. Figure 6.33  Coordinate setup for the reflectarray test.

z Main beam direction

25° 25°

x y

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Figure 6.34  The measurement setup for far‐field pattern measurement.

is linearly polarized, where note that here, the H‐plane is the xz‐plane, and the E‐plane is a plane with a tilt angle of 25o with respect to the yz‐plane. The reflectarray antennas were measured in an anechoic chamber, and a photo of the test setup is given in Figure 6.34. The reflectarrays are designed for the operating frequency of 13.5 GHz. All reflectarrays have the same square aperture measuring 400 × 400 mm2. The element spacing is set to 11.2 mm, which corresponds to 35 × 35 elements on the aperture of the reflectarray. A 1.58 mm thick Taconic TLX‐8, with ɛr = 2.55 and tan δ = 0.0019 is used for all designs. The planar surfaces lie in the xy plane and is illuminated by a linear‐polarized horn located at (−175 mm, 0, 375 mm). The phase on the reflectarray apertures are designed to direct the main beam to θ = 25°, φ = 0° with respect to the coordinate system shown in Figure  6.33 [13]. Photos of the three reflectarray prototypes are given in Figure 6.35. Comparison of the calculated and measured radiation patterns in E‐plane and H‐plane are presented in Figure 6.36 and Figure 6.37, respectively. While the agreement between measured and simulated results is quite reasonable for all designs, note that in comparison, the cross dipole shows the largest discrepancy. This is primarily attributed to the limited accuracy of the fabrication process, and the highly non‐linear slope of the reflection phase curve for cross dipoles. In other words, square patches or loops generally perform better than cross dipoles, if sufficient fabrication precision cannot be realized for the given design. The measured co‐ and cross‐polarization performance in both planes are also given in Figure 6.38 and Figure 6.39, respectively. It can be seen that in all cases, the cross‐polarization levels are below −25 dB in E‐planes, and even less in H‐planes. In  comparison, however, all designs perform very similar in terms of the cross‐ polarization level.

Reflectarray Design Examples

(a)

(b)

(c) Figure 6.35  Photographs of fabricated reflectarray prototypes using variable size elements: (a) square patch, (b) cross dipole, and (c) square loop.

It is very interesting to observe the bandwidth performance of these reflectarrays. Measured gain and efficiency as a function of frequency are given in Figure 6.40, where the results clearly demonstrate that the reflectarray using square patch elements yields the best performance in terms of bandwidth. This result agrees well with the discussions in Chapter 5. In summary, it appears that in comparison among these three types of elements, despite the fact that the square patch element has the smallest design freedom (only one design parameter, which is the patch length), it outperforms both other elements in terms of both fabrication sensitivity and bandwidth.

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Figure 6.36  Comparison of the measured and simulated E‐plane patterns at 13.5 GHz for reflectarrays using different elements: (a) square patch, (b) cross dipole, and (c) square loop.

Reflectarray Design Examples 0

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Figure 6.37  Comparison of the measured and simulated H‐plane patterns at 13.5 GHz for reflectarrays using different elements: (a) square patch, (b) cross dipole, and (c) square loop.

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Figure 6.38  Measured E‐plane radiation patterns of the reflectarrays at 13.5 GHz with different elements: (a) square patch, (b) cross dipole, and (c) square loop.

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Figure 6.39  Measured H‐plane radiation patterns of the reflectarrays at 13.5 GHz with different elements: (a) square patch, (b) cross dipole, and (c) square loop.

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Figure 6.40  The measured performance of reflectarrays consisting of different element shapes as a function of frequency: (a) gain, (b) efficiency.

­References 1 C. J. Sletten, Reflector and Lens Antennas: Analysis and Design Using Personal Computers,

Artech House Inc., 1988.

2 Y. Rahmat‐Samii, “Reflector antennas,” in Antenna Handbook: Theory, Applications,

and Design, Y. T. Lo and S. W. Lee (eds), Van Nostrand Reinhold, 1988.

3 M. E. Cooley, and D. Davis, “Reflector antennas,” in Radar Handbook, 3rd Edn, Mc­Graw‐

Hill, 2008.

4 A‐Info. Website. Data sheet available at: www.ainfoinc.com/en/pro_pdf/new_products/

antenna/Standard%20Gain%20Horn%20Antenna/tr_LB‐62‐15.pdf (accessed July 2017).

5 ANSYS HFSS, ANSYS Inc., 2014. Website available at www.ansys.com/ (accessed

July 2017).

6 W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, 3rd Edn, Hoboken, NJ:

John Wiley & Sons, Inc., 2012.

Reflectarray Design Examples

7 C. A. Balanis, Antenna Theory: Analysis and Design, 3rd Edn, Hoboken, NJ: John Wiley &

Sons, Inc., 2005.

8 D. M. Pozar, S. D. Targonski, and H. D. Syrigos, “Design of millimeter wave microstrip

reflectarrays,” IEEE Trans. Antennas Propag., Vol. 45, No. 2, pp. 287–296, Feb. 1997.

Ansoft Designer, ANSYS Inc., 2014. 9 10 NSI MI website. Website. [Online] www.nearfield.com/products/NearFieldSystems.aspx

(accessed July 2017).

11 M. J. Franco, “A high‐performance dual‐mode feed horn for parabolic reflectors with a

stepped‐septum polarizer in a circular waveguide,” IEEE Antennas and Propag. Magazine, Vol. 53, No. 3, pp. 142–146, 2011. 12 A. Yu, Microstrip Reflectarray Antennas: Modeling, Design and Measurement, PhD  dissertation, Department of Electrical Engineering, University of Mississippi, Oxford, MS, 2010. 3 Y. Mao, Broadband Approaches and Beam‐Scanning Techniques for Reflectarray 1 Antennas, PhD dissertation, Department of Electrical Engineering, University of Mississippi, Oxford, MS, 2014.

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7 Broadband and Multiband Reflectarray Antennas The reflectarray is inherently a narrowband antenna [1], [2]. With a typical design, the bandwidth of a reflectarray antenna is only a few percent, depending on the element selection, the size of its aperture, and f/D ratio. This basic limitation has stimulated ­engineers to find solutions to improve the bandwidth of reflectarrays and various broadband techniques have been developed over the years. Moreover, techniques have been proposed to provide coverage to two or more widely separated frequencies. In this chapter, we will present the design methodologies that have been developed for broadband and multiband reflectarrays.

7.1 ­Broadband Reflectarray Design Topologies 7.1.1  Multilayer Multi‐Resonance Elements As discussed in Chapter 2, a simple and practical way to obtain a progressive phase shift of the reflected field is by varying the resonant length of the patches [3], [4]. This technique allows simple manufacturing, and it produces lower cross‐polarization levels than the stubs of different lengths attached to the radiating patches. To design a reflectarray, the phase of the reflected wave should have a progressive variation over the whole surface, essentially implying that a full phase cycle should be used for the reflection coefficient. The phase range obtained by varying the resonant length of a patch depends on the substrate thickness, that is, the smaller the thickness, the greater the phase range. For a thickness smaller than one‐tenth of a wavelength, a phase range larger than 300 can be obtained, which allows for a practical design [5]. However, due to the narrowband behavior of microstrip patches, the reflection phase versus length is highly nonlinear, and shows a high slope near resonance and very slow variations near the extremes. The reflection phase is also very sensitive to frequency variations near resonance. A schematic view of variable size patch elements for reflectarrays and the reflection phase response of the elements as reported in [5] is given in Figure 7.1. As a consequence, the phase distribution on the surface of a reflectarray with patches of different sizes changes with frequency and reflectarrays are limited to narrowband applications. It is clear that a smoother phase variation can provide a wider bandwidth. This can be achieved by increasing the dielectric substrate thickness, but in that case, the total phase range will be reduced, making the elements impractical. A solution to this Reflectarray Antennas: Theory, Designs, and Applications, First Edition. Payam Nayeri, Fan Yang, and Atef Z. Elsherbeni. © 2018 John Wiley & Sons Ltd. Published 2018 by John Wiley & Sons Ltd.

Reflectarray Antennas

a1 h

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180

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6

8

10 Patch side a1(mm)

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Figure 7.1  (a) Single‐layer reflectarray with patches of variable size. (b) Reflection phase as a function of patch size. Reprinted with permission from [5].

problem is by stacking two or more patches; hence, a multi‐resonant behavior is obtained and the phase range can be several phase cycles. Then, the dielectric substrate thickness of each layer can be increased to obtain a smoother and more linear phase variation, alleviating the reflectarray bandwidth limitation. A schematic view of a double‐layer variable size patch elements for reflectarrays and the reflection phase response of the elements as reported in [5] are given in Figure 7.2. Note that the reflection phase shows a significantly smoother slope and a larger phase range in comparison with the single‐layer design in Figure 7.1. To demonstrate the broadband behavior of this element, a circular two‐layer reflectarray of 406 mm diameter was designed and built using ROHACELL 51 as separators for the center operating frequency of 11.95 GHz. The arrays were cut from a cooper clad Kevlar skin using a cut plotter and the rest of the material removed. After manufacturing, tolerance errors of 0.2 mm were measured. A photo of the fabricated prototype along with the measured radiation patterns across the band is given in Figure 7.3. Note that the pattern is fairly stable across the entire band, demonstrating a good bandwidth response. The antenna has a peak gain of 31 dBi and gain variations less than 1.5 dB across 11–13 GHz.

Broadband and Multiband Reflectarray Antennas First array First separator Second array Second separator

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Patch side a2(mm)

(b) Figure 7.2  (a) Double‐layer reflectarray with patches of variable size. (b) Reflection phase as a function of patch size. Reprinted with permission from [5].

In summary, by using stacked arrays of patches, a smooth phase variation with a phase range greater than 360 can be achieved, which allows improved bandwidth and reduced sensitivity to manufacturing tolerances. Several other designs of multilayer reflectarrays have been developed over the years [6]–[9], demonstrating the unique advantages of this broadband methodology. 7.1.2  Single‐Layer Multi‐Resonance Elements The multilayer element design introduced in the previous section can indeed increase the bandwidth of reflectarray antennas; however, it is quite desirable to realize broadband designs with a single‐layer configuration. The reason is that it is less expensive to manufacture and also it does not face problems associated with the precise layer alignment that occurs in its multilayer counterpart [10]. Note that the precise alignment of multilayers can be a serious problem in large size inflatable reflectarrays [11].

181

Reflectarray Antennas 0 –5 Radiation Patterns(dB) Ex

182

–10

11.00 GHz 11.95 GHz 13.00 GHz

–15 –20 –25 –30 –35 –40 –45 –90

(a)

–60

–30 0 30 Theta (degrees)

60

90

(b)

Figure 7.3  (a) A two‐layer reflectarray prototype. (b) Measured radiation pattern of the antenna across the band. Reprinted with permission from [5].

As mentioned earlier, the difficulty with the typical single‐layer microstrip reflectarrays is that a slower phase gradient is accompanied by a reduced phase range, making it impractical. The multilayer approach introduces multiple resonances, which was the key in increasing the phase range, allowing one to utilize thicker substrates and ultimately improve the bandwidth. Hence if one can introduce multiple resonances with a single‐layer configuration, it could also be possible to use thicker substrates and improve the bandwidth. A detailed study on this design methodology was reported in [10], which is summarized here. Multiple resonance elements in the form of multiple rings of square or circular shapes were studied for broadband single‐layer reflectarrays, and the phase characteristics of a unit cell containing these uniplanar elements were compared with those of a unit cell containing stacked two‐layer patches. The geometrical models of these multi‐resonance elements are given in Figure 7.4. Both elements have several design variables that can be utilized to achieve an almost linear phase response as a function of ring size. For the double‐square ring, the size of the inner ring was set to 0.9 the size of the outer one. The width of the two rings was set to 0.025 of the outer ring size. Two different substrates were studied. The first substrate had a thickness of 1.57 mm and a dielectric constant of 3.2. The second substrate had a thickness of 3.175 mm and a dielectric constant of 2.2. The reflection phase response of this element is given in Figure 7.5(a), where it can be seen that the phase range obtained for the double‐square ring is approaching 700°. The use of substrate with larger thickness and smaller dielectric constant results in lower slope and more linear response. This is at the expense of slightly reduced phase range of about 650°. Similar trends were observed with double‐circular rings on the same substrates. For this case, the inner ring radius was 0.8 of the outer ring radius while all the rings have a width of 0.08 of the outer radius. Similarly, the obtained phase range for the thinner substrate is approximately 700° while that for the thicker one is about 650°. It is also important to note that in comparison with the square‐ring element, the slopes of the

Broadband and Multiband Reflectarray Antennas

W1

W1 W2 W2

L1

L2

L2

2R2

L1

2R1

Figure 7.4  Geometries of double‐square and double‐circular rings. Reprinted with permission from [10].

Phase of reflected wave (deg)

100

t=3.175: Er=2.2, L2=0.9L1, W=0.025L1 t=1.57: Er=3.2, L2=0.9L1, W=0.025L1

0 –100 –200 –300 –400 –500 –600 –700

3

4

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6

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9

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Size of Outer Ring L1 (mm)

(a)

Phase of reflected wave (deg)

100

Er=2.2, t=3.175mm; R2=0.8R1, W=0.08R1 Er=3.2, t=1.57mm; R2=0.85R1, W=0.08R1

0 –100 –200 –300 –400 –500 –600 –700

2

3 4 5 6 Radius of Outer Ring R1 (mm)

7

(b) Figure 7.5  Phase response of the double‐ring elements as a function of outer ring size: (a) double‐square, (b) double‐circle. Reprinted with permission from [10].

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Reflectarray Antennas 0

Patch #1 3 mm Foam Patch #2 3 mm Foam Ground

Rings 6 mm Foam

Phase of reflected wave (deg)

184

–100 –200 –300

–500 Ground

(a)

Stacked patches R2 = 0.6R1, W = 0.2R1 L2 = 0.5L1, W = 0.125L1

–400

2

4

6 8 10 Size of Outer Ring (mm)

12

(b)

Figure 7.6  (a) Configurations of composite substrates. (b) Phase responses of double‐circular rings and double‐square rings compared to that of the stacked two‐layer square patches. Reprinted with permission from [10].

circular rings are lower, which is also a trend that is observed with square and ­circular patch elements. In designing double‐ring elements, one has the choice of the sizes of the two rings, their aspect ratio, and their widths. For dual‐polarized operation the aspect ratio is unity, and thus, the designer has three degrees of freedom, namely, the relative size of the inner ring and the widths of the two rings. Some of these parameters can be adjusted to linearize the response. The procedure is similar to staggering two resonance responses, where the size ratio is responsible for the separation between the resonance frequencies, and the ring widths mainly control the widths of the two responses or the local slope of the phase response. Furthermore, thicker substrate and lower dielectric constant facilitate an easier achievement of linear phase response. These parameters were utilized in [10] to demonstrate a linear phase response for double‐ring elements, which was also compared with the multilayer technique. The geometrical models of the elements along with the reflection phase response are given in Figure 7.6, where it can be seen that the single-layer double‐ring elements perform almost identical to the double‐layer design. This single‐layer multi‐resonance approach has received considerable attention and several reflectarray designs of these elements have been reported in literature [12]–[16]. 7.1.3  Sub‐Wavelength Elements Traditionally, the reflectarray phasing elements are designed with unit‐cell sizes around half a wavelength. Advances on metamaterials [17], however, revealed that similar reflection phase response can also be realized using sub‐wavelength elements. A patch element placed in a lattice with half-wavelength spacing can be self‐resonant. In other words, it can exhibit resonance as a single element. A single electrically small patch, however, will no longer exhibit resonance and typically cannot be used as a radiator. For reflectarray designs, as long as a sufficient phase range is obtained with the element, smaller unit‐cell periodicities can be used for the element. This is due to the fact that the

Broadband and Multiband Reflectarray Antennas

adjacent elements in the periodic lattice of the reflectarray give rise to a coupled resonance behavior. This coupling mechanism between adjacent elements provides a useful resonance that can be utilized to phase the aperture of the reflectarray [18]. The design approach is basically identical to the traditional method, which makes it rather simple to implement. The only difference is that the smaller lattice size results in a larger ­number of elements. The first mention of sub‐wavelength elements for reflectarrays was given in [19], using the term “artificial impedance surface,” and it was shown that these elements can provide a wider bandwidth than traditional half‐wavelength elements. While some gain loss was observed, primarily due to a reduction of the phase range of sub‐wavelength elements, a substantial improvement in reflectarray gain bandwidth was observed. To gain an understanding in the operating mechanism of sub‐wavelength elements, we compare the reflection properties of reflectarray phasing elements with periodicities smaller than half‐wavelength. The variable size square patch is used, and the only difference is the selection of element periodicity. In particular, λ/2, λ/3, and λ/4 are selected as the periodicities for this study. Although smaller periodicities such as λ/10 can be selected in the analysis and design, it will bring difficulties to the fabrication tolerance and increase the fabrication cost, which will be further discussed later on in this section. A 20 mil Rogers 5880 substrate (εr = 2.2) is used here for this study. The reflection coefficients versus patch size at the design frequency (32 GHz) for normal incidence are given in Figure 7.7. Comparison between the reflection magnitudes of the elements shows that as the unit‐cell size decreases, the resonance moves toward smaller patch sizes, indicating a stronger coupling effect between adjacent patches. Furthermore, note that the element losses also decrease. While the reflection coefficients for all the elements in this study are larger than 98%, mainly due to the low‐loss properties of the substrate, sub‐wavelength elements can potentially reduce the element losses [21], [22]. As pointed out earlier, comparison between the reflection phases of the elements also shows that as the unit‐cell size decreases the phase range is reduced and the slope inclination becomes more vertical, essentially increasing the quantization error in the fabrication process. In general, this is one of the major challenges in sub‐wavelength designs. With respect to the phase reduction behavior of the sub‐wavelength elements, it is important to note the effect of the gap size between the patches. In the unit‐cell analysis here, a periodic boundary condition is being used to account for the coupling between the unit‐cell elements. The coupling between the elements is a function of the patch size and the gap between the patches. For a fixed unit‐cell size, the coupling between the patch elements increases by reducing the spacing between them. However, if the size of the unit cell is reduced, a closer spacing between the patch elements is required to achieve the same level of coupling for larger unit cell sizes. Consequently, this would mean that smaller unit‐cell sizes with the same gap size would have a weaker coupling between the elements, which would reduce the phase range versus patch size for these elements. More discussion on the phase range of sub‐wavelength elements and the effect of gap size between the patches will be given in a subsequent section. Finally note that the results presented here are under normal incidence approximation. In general, the reflection characteristics are angle dependent and oblique incidence needs to be considered. However, as mentioned in earlier chapters, it has been shown that normal incidence can present good approximations for certain elements

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Reflectarray Antennas 150

Ansoft Designer (λ/2) Ansoft Designer (λ/3) Ansoft Designer (λ/4) Ansoft HFSS (λ/2) Ansoft HFSS (λ/3) Ansoft HFSS (λ/4)

100

< Γ (degrees)

50 0 –50 –100 –150 –200

0

1

(a)

2

3

4

5

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Patch Width (mm)

1 0.99 0.98 Γ

186

Ansoft Designer (λ/2) Ansoft Designer (λ/3) Ansoft Designer (λ/4) Ansoft HFSS (λ/2) Ansoft HFSS (λ/3) Ansoft HFSS (λ/4)

0.97 0.96 0.95

(b)

0

1

2 3 Patch Width (mm)

Figure 7.7  Reflection coefficients versus patch size at the center frequency 32 GHz for λ/2, λ/3, and λ/4 unit‐cells with normal incidence: (a) phase, (b) magnitude. Reprinted with permission from [20].

with incidence angles up to 30o. Moreover, a study on the properties of sub‐wavelength elements was given in [23], which showed that sub‐wavelength elements provide a reduced angular sensitivity over their resonant counterparts, making the normal incidence approximation a more valid choice in these designs. The element studies presented so far gives us a basic understanding of the element design methodology, however, fail to provide a solid explanation on bandwidth characteristics. From an element analysis viewpoint, operating off‐self‐resonance is considered to be the reason for the broadband behavior of sub‐wavelength reflectarray elements. As discussed earlier in Chapter  5 the key to understanding the bandwidth behavior of reflectarray elements is the phase analysis as a function of frequency, thus, here we analyze the frequency response of half‐wavelength and sub‐wavelength reflectarray elements. Consider a case where two elements on the aperture are required to have a 90° relative phase difference. Using the element data in Figure 7.7, for a zero‐ degree reflection phase, the patch size will be 2.69, 2.41, and 2.04 mm for λ/2, λ/3, and

Broadband and Multiband Reflectarray Antennas

λ/4 designs. When a 90° reflection phase is required, the patch size should be 2.41, 2.1, and 1.81 mm for λ/2, λ/3, and λ/4 designs. Figure 7.8(a) shows how the reflection phases of these 0° and 90° elements vary with frequency. It is observed that λ/2, λ/3, and λ/4 designs have different frequency behaviors for the element reflection phase. When the frequency changes, the closely spaced elements show a smaller phase variation with frequency. As discussed previously, it is the frequency behavior of the phase difference that determines the reflectarray bandwidth. Thus, the phase differences between these 0° and 90° elements are calculated, and the phase error curves of the λ/2, λ/3, and λ/4 designs are obtained and plotted in Figure 7.8(b). It is clear from this figure that the closely spaced elements have a smaller phase error over the frequency than the half‐wavelength elements. Similar studies have

150